A full school year with virtual classes, virtual office hours, virtual collaboration, and virtual meetings seems insane to think about. Since this school year will always be remembered for its peculiarity, I wanted to go through a bunch of interesting yet academic-related things I learned during this school year. 1. How a DRAM cell works.One of the most interesting takeaways from this year was how computer memory operations actually worked. DRAM stands for Dynamic Random Access Memory. It is a type of random access memory. What makes random access memory great is the ability to fetch/write to any 'random' memory location at one time. If the computer needs to fetch some data from memory, it can access it by simply using the instruction's physical address (the address that tells the computer where the instruction is in memory). Dynamic RAM is known as 'dynamic' due to the nature of a DRAM 'cell'. Each DRAM 'cell' stores a bit of data, by utilizing 1 transistor and 1 capacitor. The diagram below shows what a DRAM cell looks like. The bit being stored in the cell depends on the presence of charge in the capacitor. A capacitor is essentially an electrical component that is made up of two metal plates separated by a dielectric (an insulator that can be polarized). A capacitor has some charge if one plate gains positive charge, and the other plate builds up negative charge. If the capacitor in the cell is charged, then the bit stored is a '1'. If the capacitor has no charge, the bit stored is a '0'. So how does this all work? How does a read/write operation to memory happen when each bit is stored in the configuration above? There is a pass transistor and a capacitor in each cell. The transistor essentially acts as a switch; if the word line is ON, the transistor connects the capacitor to the bit line. If the word line is OFF, the capacitor is not connected to the bit line. When a memory write occurs, the bit line is set to the bit to be stored. Then, the word line is also asserted. If the bit being stored is a '1', the bit line is turned ON, and the pass transistor connects the capacitor to the bit line, which charges the capacitor. Once the capacitor is fully charged, a '1' is stored in the DRAM cell. If the bit line is '0', there is no charge in the capacitor and a '0' is stored. When a memory read occurs, the bit line is set to half of the 'ON' voltage. There is a set voltage (Vcc) that represents a '1' ON voltage. By setting the bit line to half of the voltage, we are performing a reading operation (trying to read the charge in the capacitor). After the bit line is set to Vcc/2, and the word line is asserted, the magic happens. If the capacitor is charged, the capacitor will start to discharge (since there is a potential difference between the capacitor voltage and the bit line voltage). If the capacitor is not charged, the capacitor will start to charge, using the current supplied from the bit line. In both of these cases, the bit line's voltage will slightly change by a small amount below or above the Vcc/2. A special component called a sense amplifier senses these changes, and reads the bit value depending on the changes. If there is a positive change, the value is a '1', else value is a '0'. However, after reading the values, the capacitor's charge changes--therefore, the read operation is a destructive read. Therefore, we need a refresh circuit to refresh and pre-charge the capacitors, to keep the original values. This is why DRAM is a dynamic RAM--it requires these refreshes. A DRAM array is made up of a bunch of DRAM cells put together, which then make up DRAM banks. This makes up DRAM chips, with then make up ranks. A DIMM module consists of a bunch of DRAM chips. DRAM is useful because it provides more capacity, at a lower cost. I didn't at all know about how memory looks like at a micro transistor level of things, so this was an eye-opening lesson for me. 2. Dijkstra's algorithm...!One of the coolest theorems used in many real life applications today is the Dijkstra's theorem. The gist of the algorithm has to do with finding the shortest path between two nodes/locations in a given graph. A very interesting application of this algorithm is finding the shortest route in a given network. For instance, packet routing from one computer to the destination computer involves finding the shortest path between these two 'nodes'. Dijkstra's algorithm is prevalent in many applications today. 3. Working backwardsWorking backwards is a very useful technique that can be used for many problems. I think this is a really important technique! Congrads to the class of 2021! Thank you! Aarushi
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Complex numbers: One of the main mathematical concepts when it comes to describing physical systems, such as signals, and circuit analysis! Let's delve deep into what complex numbers are and more about its applications. What makes a number "complex"? And what is the "real" definition of a "real" number? (ha!) real vs. imaginary numbersReal numbers consist of integers, whole numbers, positive or negative, rational numbers and irrational numbers (or a combination of those!!). Imaginary Numbers on the other hand, consist of numbers with a property where it's square is negative. An imaginary number, i, represents the square root of -1, where i^2 = -1. NOTE: Sometimes imaginary numbers are denoted with a j. a complex numberA complex number is a number that's made up of a real part, and an imaginary part (that's what makes it 'complicated'). A complex number (usually denoted as z), is written in the form: a + bi, where a represents the real part of the complex number, and b represents the imaginary part of the complex number z. Note that a and b are real numbers, but since b is being multiplied by an imaginary number i, b is the imaginary part, or Im(z). A really interesting part about a complex number is that it can be represented graphically, just like how we plot coordinates in a Cartesian plane. (Like (2,4) on the Cartesian plane for example). The difference is we plot complex numbers in the Complex Plane, where the vertical axis is the imaginary axis (where imaginary numbers are plotted - i, 2i, 3i..), and the horizontal axis is the real axis (where real numbers are plotted - 1, 2, 3..). To plot a complex number a + bi, you would essentially move a steps horizontally (horizontal real axis) from the origin, and then b steps vertically or perpendicularly (vertical imaginary axis). The the magnitude of z, represents the distance from the origin [(0, 0)] to the complex number z's point. The distance is calculated by sqrt((difference in x axis)^2 + (difference in y axis)^2). So it's calculated by adding the squares of a and b and taking the square root of the sum. (Just like how you would calculate the distance in the cartesian plane or in a map, for example). For any real number, a one -dimensional number line would satisfy to plot the number. But for a complex number, we would require a 2 dimensional plane to plot it's imaginary part and real part. The really interesting part of this complex number representation is the angle, or the phase. To calculate the angle, we can use the distance between the origin and the point, r! We can say that a, the horizontal portion is equal to the cos(of the phase angle) * r. (Since cosine is adjacent over hypotonuse) and b the vertical portion is equal to sin(of the phase angle) * r. The tan of (the phase) is then equal to b/a! Since the r's cancel out when dividing. So you can then find the phase of the complex number (also called argument of z) by finding inverse tan of b/a. euler's formulaThe Euler's Law relates the these cosines and sines we see above with a complex exponential using a revolutionary formula! The Euler's formula states that: So essentially the component to the left of the equation is a complex exponential function, e raised to the power of (imaginary number * a phase/angle x). This is equal to the cos(x) + imaginary number i * sin(x). What makes this formula so intriguing is that there is a way to represent sines and cosines using complex exponentials, which can be very useful in a lot of applications. applicationsThere are numerous applications of complex numbers and complex exponentials! One of the applications is in Signals and Systems/Processing, where you can express periodic signals as a bunch of sines and cosines (sinusoidal functions in Fourier Analysis). These sinosoids can also be expressed as complex exponentials, since in LTI systems (Linear and Time Invariant systems), complex exponential inputs can make calculations much more efficient/easier! (These are also called eigenunctions of LTI systems!) Thank you!
WHEW. WOOOOOOOOOOWIEE. As this unusual yet unique semester comes to an end, I wanted to look back at the things I've learned as a sophomore. It's insane to think that Junior Year is approaching, and I think the best way to commemorate this school year is to go through the lessons/the things I learned about, especially because of how the method of learning has changed because of these challenging times. Also, to the class of 2020, CONGRADS!!! I wish you all the best for your future, and I know you are going to rock it! So I learned lots of things from all my classes (Of course!!), but I think my favorite class by far was the Systems and Signals class. The concepts I learned in my signals class were so applicable to real life scenarios! For instance, when you send a message to another device in your WiFi, an electronic signal transmits the bits of info (data) to the router which transmits It to the receiver! Signals are everywhere! So, these were some of the key points I felt were pretty 'special' to me this semester: 1. Frequency Domain vs. Time DomainUnderstanding the two different types of domains in signals and systems is important and very interesting! The time domain graph shows how a particular signal (waveform, sinusoid) changes over time. "Frequency" is essentially the number of times a particular instance of a signal occurs. So, the frequency time domain graph shows how much of the signal is in a frequency band over a range of frequencies. 2. Lesson on Fourier transformThis topic was very intriguing, and I wanted to dedicate to it in a separate post (coming soon!!). 3. Thinking "outside" of the box ProblemsThere were many problems in my classes that were challenging to solve, but one particular problem that comes to my mind was in my Digital Logic Design class. We were asked to design a grading system, using only a certain number logical components (a 2-by-1 MUX, 2x4 Decoder, and 2 Comparators). Since there was a restriction as to how many digital logical components you can use, I couldn't just use as many components as I wanted, wherever I want. I needed to understand the function of every component and place them in the appropriate place. It was a tough one, since I had to essentially "experiment" with every possible combination placement of these components. I only had two comparators, and I had to decide whether the input number was a grade A, B, C or D. It was a tricky one, and I finally had to see the answer because my brain was literally about to explode lol. But as soon as I saw the solution, I finally had my AHA moment and realized that I could use the MUX as a component to restrict the possible range of our input. Thinking outside the box is all about understanding the function and using it where it is the most applicable (especially if you don't have that many resources). 4. Real Life Scenario ConnectionsReal Life Scenarios. Super important, especially in my two EE classes. Digital Logic Design is all about the digital components that exist in our computers today, and we essentially looked into the several smaller components that make up these important parts. It's just insane to learn about what an SRAM or DRAM (new posts, maybe?!) looks like, since we have been hearing these words forever but I never really understood what they meant (or how they looked liked) UNTIL NOW!! (no flex) 5. ReflectionsI'd like to end it off by saying, its been a great and CRAZY year. I know it sounds cliché, but sophomore year was a pretty valuable year to me. I actually learned A LOT, and not just about academics/electrical engineering/computer science, but about life. How important it is to be a be a good citizen, stay home, help others, and help yourself of course. How life can give you the most unexpected turns, and how you should just keep on going, and keep on moving. I hope you enjoyed your year as a freshman/sophomore/junior/senior, and once again thank you for reading! I wish you the best, and GOOD LUCK!
Thank you, Aarushi Ramesh :) ***During the COVID-19 outbreak, it is important to remember not to panic, but to be precautious and stay safe. Please be sure to wash your hands thoroughly and properly, and practice social distancing at this time.*** Stay safe we can fight this ❤️ I've actually never even been amazed at exponential growth and its graph until I watched this YouTube video of how it is related to epidemics: www.youtube.com/watch?v=Kas0tIxDvrg&t=8s. I highly encourage you watch it, to learn about the applications of exponential growth. Exponential growth is all about increasing/decreasing growth with respect to the previous value in time. For instance, if the population (of any kind) keeps increasing by a factor of 2 every week, then the current value is 2*previous value. So if the population starts at 2, it go up to 4, 8, 16, 32, ....for every week and so on.
Obviously, there needs to be a tipping point or a stabilizing point where the growth is 'stabilized' and is set to a maximum value reach. This is where the population stays constant at a certain value (tho in real life there may be some fluctuations). This means that in real life, things can't really 'grow exponentially' forever. There has to be a point where it stabilizes due to external changes. Since things can grow exponentially only for some time (and not forever), here is a more realistic formula, or expression: f(t) = a(1+r)^tHere, the t represents the time (in months, years, weeks or days), r represents the growth rate, and a represents the initial number we started with. The 1+r represents the total constant that is being multiplied by the previous value in time, as I mentioned earlier. So in this case, it is (next_value) = (1+r)*(prev_value) Image of an exponential increasing by a rate of 0.15. However, exponential growth doesn't stay forever, so when it balances out and stabilizes to a constant value, it resembles a logistic growth curve. It means it has a carrying capacity. Since exponentials trail on to infinite values, we can't rely on this to predict population growths..etc. Image of an exponential increasing by a rate of 0.15. This collab notebook from Amita Kapoor plots the number of cases in different countries, compared to an exponential curve graph: colab.research.google.com/drive/1jJAa7QFOCMRrQu4ZAJg6Rc4wNEWF-jFn Stay safe during these times, and don't panic. There will be a stabilizing point, but in order to stabilize it soon, we need to make sure to practice social distancing.
A reasonable probability is the only certainty. - E.W. Howe The best way to describe the term "Probability" is that its a combination of very interesting concepts with 500 things to keep in mind about (ok maybe not 500 but a LOT of formulas lol). Yeah, I feel like Probability has always been the math thats "weird" and different--not in the bad way though! I think once you learn a lot about its applications in the real world, it gets very interesting and exciting! If you're interested in the probability concepts of sets, unions and intersections, I got you covered! Its on this link! If you want you learn more about Probability models and functions, this is for you! (hopefully lol) probability models & PMFsSo whenever we are considering the probability of a certain event to occur, we are also considering the total number of possible outcomes. How can we lay out the possible outcomes for a particular situation (ex. flipping a fair coin n times, or finding the number of k successes in t trials) with certain values associated with it, like for example the number '1' to represent the probability of heads to appear, and the number '0' to represent the probability of tails to appear?? (yeah that was the longest question in the world lol). In order to answer this question, we need to learn about the Probability Mass Function, aka PMF. The PMF basically maps out certain values to the probabilities (outcomes) of an experiment or situation. But what are those 'certain values'? These are called random variables, or discrete random variables. A Random Variable is a variable that assigns a value to a certain outcome of an experiment. We use a capital letter to represent the random variable of an experiment. So for example, lets say the experiment was to test 10 different circuits and see if they work (a success, denoted as s) or they need improvement (an error, denoted as e). Every observation in this case is a sequence of 10 different letters (s or e). So the sample space or the total number of possible outcomes (sequences) is 1024 (2^10). The random variable in this case, lets say K, can be the number of successful circuits in a sequence. So for a certain outcome, sssssseeee, the random variable K = # of successes = 6. Since the sequence is a set of 10 letters, the range of K has to be from 0-10. K is a discrete random variable, since the range of K can be listed. (even if it was infinitely long). goal scoring problemOk lets look at another example with pmf and random variable (RV) stuff. Suppose you are playing soccer and have two free kicks. A free kick can lead to two possibilities: a score (in the goal, denoted as S) or no score (N). What is the PMF of the random variable G, the number of free kicks scored? We can make a table to highlight the probability of the scoring a certain outcomes and its relationship with the random variable G! Since we are assuming that each outcome is equally likely, the probability of getting a goal in the first try and not getting a goal in the second try is just = 1/2*1/2. This is the same for every other outcome.
Since the random variable G has three possible values {0, 1, 2}, the probabilities of these three possible values are: P[X=0] = 1/4, P[X=1] = 1/4 + 1/4 = 1/2, P[X=2] = 1/4. One way of representing the PMF is by a plot or graph: Note: The PMF/probability is represented with a P and then a subscript of the random variable, and then a lower case version of the random variable in the brackets, because the notation says that this is "the probability of a random variable G equaling some value g (which is in the set of outcomes)". So g is an actual number, which is in the range of G, the random variable. binary symmetric channelOk this is a trickier problem, and what I really liked about this one was that it is has such a great connection with electrical engineering. So, we are sending a 1000 bits into some data channel, and the probability there is a bit error (the bit was the wrong bit received, for example 1 is a 0 or 0 is a 1) = P[error] = 0.02. So 1-P[error] = success. So the question is, what is the probability there are 10 errors? Ok so first, we need to think about how many possible combinations are there if there are 10 errors out of 1000 bits sent into the channel. In order to figure out the total number of possibilities we do 1000 choose 10 (the combinations formula). We would then use the binomial distribution (since there are only 2 possible values, 1 or 0 to find the PMF: If we plug in P for 0.02, we get about P[10 errors] ~~ 0.0055, which is a relatively low probability!
I remember those fun middle school days when my math teacher used to teach us probability by bringing in a bunch of marbles and bags to find out what is the chance you pick a blue or red or pink marble, etc. Of course, we as college/high school students are gonna learn more than that. Probability is basically the 'science' and 'math' behind the uncertainty or certainty of an event. And it is applied EVERYWHERE. defining your sample spaceWhenever you are dealing with probability problems, you always need to define your sample space. The sample space is essentially the 'space' where you describe all of the possible outcomes of the event. It is also known as the symbol omega. find the probability lawThe next step is to find your probability law. The probability law is basically the 'formula' used to assign probabilities to events of an experiment. For example, in some experiments (which we will talk about soon), the probabilities are determined by the area the events take up in the sample space. This is called a continuous probability model. In other cases, the probabilities of an event are determined by simple counting. For example, lets say this was your sample space: The orange part in the diagram is called an event. An event is just a collection of outcomes in a sample space. Lets say omega (sample space) consists of 6 equally likely elements. What is the probability you randomly choose a dot in the orange part of the board? Well, we know there are 6 different possible outcomes, because there are 6 different points. So, we basically divide the probability of the event with the total number of possible outcomes. In that case, the probability will be 3/6. So a half. Not that tricky, right? discrete uniform lawSo the example we just went over supports the discrete uniform law. The discrete uniform law says that if the sample space consists of n equally likely elements, and A, which is a subset of the sample space, consists of k elements, Probability of A = P(A) = k * 1/n = k/n Since all of the elements are equally likely to be chosen, each element has a probability of 1/n. Since A has k elements, the probability of A being chosen is k * 1/n = k/n. intro to axiomsOk, so far, so good. But what if the probability of something is negative? Is that even possible? How do we put rules and limits to these probability values? This is where axioms come in. Axioms are certain properties that probabilities need to follow in order to be valid. These are the three main probability axioms:
The first axiom is the non-negativity axiom. This axiom tells us that all probabilities have to be greater than or equal to 0. They CANNOT be negative. A probability of 0 indicates it is uncertain to happen, whereas a probability of 1 tells us that it is certain. The next axiom tells us that the probability of the sample space, or omega is 1. This makes sense because the sample space consists of all the possible outcomes. So it is absolutely certain of the event is going to happen, since it is in the sample space. The last axiom is famously known as the additivity axiom. The additivity axiom tells us that if there are two events that are disjoint (we will talk about this soon), then the probability of both the events together is the sum of the individual properties. I know that was a lot of info, so lets break it down: Look at the above diagram. Lets say we have two events in our sample space, Event A (highlighted in yellow), and Event B (highlighted in red). The intersection of these two events is the orange part. You denote the intersection of two events like this: A ∩ B. The intersection of two events means that there is a chance A and B can happen at the SAME TIME. The union of two events means there is a chance A or B can happen. You denote the union of two events like this: A ∪ B. So in the above example these two events were NOT disjoint. Disjoint events are two events or more where no intersection occurs. When both the events don't have an intersection they are disjoint. So, In this case, Events A and B are disjoint. So, back to the additivity axiom. The additivity axiom says that, if the intersection of two events is 0 (disjoint), then the P(A ∪ B) = P(A) + P(B). So this is saying that the union of these two events is the sum of the individual probabilities. Which makes sense because they are both separate events and we want to find whats the probability we land in event A or B.
Anyway, probability is a tricky but really cool concept to learn. Good luck! Fourier Series. This was the super annoying memorizing-the-formulas topic in my class, but it was actually one of the most interesting math concepts ever. So what is the Fourier series? What's the significance of it? For that we first have to understand what a periodic function is. A periodic function is a function where T > 0, and f(x+T) = f(x) for every value of x. The T is the period of f(x). An example of a periodic function is sin(x) and cos(x), which have a period of 2π. Fourier series is essentially a way to expand this periodic function to an infinite series involving a bunch of sines and cosines. So how do you derive the Fourier series of a periodic function? Let p > 0 and f(x) be a periodic function with period 2p, within the bounds of (-p, p). The Fourier series of f(x) is: where the a of n and a of 0 and b of n are Fourier coefficients ALSO, there are two things to keep in mind: assuming x is an integer,
the left graph shows the original f(x) and the right graph shows Fourier series estimation graph. When you graph the certain number of n terms of the fourier series, you will get a close approximation. This series is very similar to Taylor series, except Fourier series also works with discontinuous functions as well. Fourier sine seriesYou can obtain the Fourier sine and cosine series from the general formula. For the Fourier Sine Series, we assume that f(x) is an odd function, which means that f(-x) = -f(x). If thats the case, then the a of 0 and a of n terms become zero because an odd function (f(x)) multiplied by an even function (cos(nπx)) = an odd function. An interval from -p to p over an odd function is 0. an example of an odd function over an interval (-p, p). both the areas cancel each other out, which evaluates the integral to 0. Since a of n and a of 0 are both equal to 0, the function evaluates to the general Fourier series with just the b of n Fourier coefficient. Fourier cosine seriesYou obtain the Fourier cosine series when you assume that f(x) is an even function; which means that f(-x) = f(x). Since f(x) is an even function, we know that the b of n term in the general Fourier series equation is 0 because an even function (f(x)) times an odd function (sin(nπx)) is equal to an odd function. The integral from (from -p to p) of an odd function is always zero since the areas cancel each other. Therefore, with only the a of n an a of 0 terms, the Fourier series becomes the Fourier cosine series (with only cosines).
I never actually really understood the significance of eigenvalues and how it's applied visually until my Diff Equations class was over, which is not the best time to figure it out, but at least it'll help me for my future classes, lol. And, its actually pretty cool too!! Honestly, I think the time when I got super interested in learning about why these eigenvalues existed was when this scene came up in the Avengers Endgame. When Tony Stark mentioned the word "eigenvalue" I was like, "OMG, OMG I actually kinda know this...kinda...that word is very familiar to me so it counts." LOL, yup, this is when I was like hmm I have to actually understand an eigenvalue's significance and application. So wow, that happened and I never would have expected an Avengers movie to squeeze in a quick lesson in Diff Equations and Linear Algebra. Anyways, getting back to the point, what is an eigenvalue? And what are these used for? For example, when you solve for an IVP (initial value problem) with matrices, you solve for the eigenvalues first by finding the determinant, and then you solve for the corresponding eigenvectors (from the eigenvalues), and then plug in the initial value equation to the general solution to find the value of the constants. That sounds terrible, but it's actually not too bad. It's basically just solving for linear equations except in a matrix you have to find the determinant. That's one way of using eigenvalues. But what are they? and what's their real world application? Eigenvalues and their corresponding eigenvectors summarize matrix data. Eigenvectors are vectors whose direction is not changed when some linear transformation is applied to the vector. For instance, The red vector is an eigenvector since it never changes, even after a linear transformation has been applied. These vectors define the matrix of the transformation (scaling). For any matrix A (n x n square matrix), x, (a n x 1 vector) is an eigenvector of this matrix if the product of Ax is proportional to the product of x * eigenvalue. where x is the eigenvector, A is the matrix, and the lambda is the eigenvalue. When you solve for lambda using this formula, you will end up with: For this to be equal to 0, A has to equal lamda times I, where I is an identity matrix with the same dimensions as a, and lamda (eigenvalue) is just a scale factor. However, we are assuming that x is not a null vector, which means to satisfy the equation, A - lambda * I can't have an inverse. A matrix that is non-invertible has a determinant of 0. Therefore, we can conclude that, And we just use this equation to solve for eigenvalues of any n x n matrix.
Eigenvalues and eigenvectors of a certain matrix have tons of real world applications such as image compression, clustering in data science, predictions and page rank algorithms. The Cantor's Disappearing Table is a famous and super fun series problem regarding a table and whether it'll disappear by cutting only half of the table. Sounds incredibly absurd but it is super interesting to solve! Lets explore this problem in depth: We will start with a table, with the length L. Lets remove 1/4 of the table from the center, leaving two pieces in the sides. The remaining pieces each definitely have a length less than 1/2L. Then, lets remove 1/8 of the table by removing 1/16th from each of the remaining sides. Each of the remaining pieces have a length less than 1/4L. Next, we remove 1/16th of the desk by removing 1/64th from each previous remaining piece. Each of the remaining pieces have a length less than 1/8L. As we continuously remove this portions of the table, will the table eventually disappear? There are two ways to look at this problem. One way is focusing on the amounts removed from the table, and the other way to look at this problem is focusing on the amounts remaining. Amounts RemovedSo if we continuously remove the portions of the table in this pattern, the series works out to be: 1/4L + 1/8L +1/16L + 1/32L + .... = (n=2 to ∞) ∑ (1/2^n)L = 1/2L. This means, that we will actually remove half of the table by following this process. I obtained 1/2L by: S = 1/4L + 1/8L +1/16L + 1/32L + .... S = (1/2)^2 + (1/2)^3 + (1/2)^4 + (1/2)^5... = (1/2)^2 + 1/2( (1/2)^2 + (1/2)^3 + (1/2)^4) = S = (1/2)^2 + 1/2(S) 1/2S = 1/4 S = 1/2 So this leads to the conclusion that only half of the table will ultimately disappear but the entire table won't. Seems really absurd, but it is mathematically proven to be correct in this way. Amounts RemaningSo if we focus on the amounts remaining, we will end up with a different conclusion: that the table does eventually disappear. Lets make a series for the remaining pieces: S = 1/2L + 1/4L +1/8L + 1/16L + .... In order to find out how much is remaining: ; lim(as n approaches ∞) (1/2)^n = (1/2)^∞ = 0. So this means that the remaining lengths will approach 0L. This means that the pieces will ultimately have a length of 0. This proves that the table eventually disappears. We can conclude that there are two different sides to this problem. It just depends on which way you choose to look at it, whether it is the remaining pieces or the removed pieces. This problem was so fun to solve and it was so interesting to prove the two different perspectives!! Good luck as always!
A sequence is a function whose domain is basically a set of positive integers. We denote these list of values with a variable and a subscript of 1, 2, 3,... all the way up to n. This shows a series of a set of numbers. 2 + (-1)^1, 2 + (-1)^2 + 2 + (-1)^3 + 2 + (-1)^4.. 1, 3, 1, 3 Sequences approaching limiting values converge, and sequences approaching a value that does not exist are said to diverge. The definition of a limit of a sequence is: lim (n approaches ∞) a(nth term) = L where L is a limit. If L exists, the sequence converges. If limit L of a sequence does not exist, it diverges. For the above example, because the sequence alternates between 1 and 3, the limit does not exist. Therefore, the above sequence diverges. Squeeze Theorem for Sequences
Monotonic Sequences and Bounded SequencesA sequence is monotonic when its terms are non-decreasing or non-increasing. To determine whether a sequence is monotonic, simply check whether the terms in the series are increasing or alternating. If it is changing between two values, it is not monotonic. A sequence is bounded above when the real number M is greater than the sequence for all numbers. M is the upper bound in this case.A sequence is bounded below when the real number M is less than the sequence for all numbers. M is the lower bound in this case. When a sequence is bounded above and below, it is bounded.
Therefore, from that, an - L < ϵ which means that {an} converges to L. It is super interesting to delve into the depths of series and their applications in the real world too. Explore on! Good luck!
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