I probably needed to repeat that. I know, though it sounds ridiculous, the sum of all natural numbers leading up to, I suppose infinity is proven to be -1/12 by many of the greatest mathematicians of all time. How about we become a mathematician and prove it once again? Instead of using complicated formulas and other ways of solving this puzzle, we can use a simpler way to prove it. I also wanted to talk about many other ways of solving this and mathematician's efforts to proving this as well. (especially from my favorite and highly regarded mathematician, Srinivasan Ramanujan). I saw this phenomenal fact in a youtube video on proving this (https://www.youtube.com/watch?v=w-I6XTVZXww), and wanted to share it with you all. Let's prove this infinite series in a simple way: Step 1. Write the first sum.First, we are going to write three different sums, labelled as S1, S2, and S3. The last sum, S3 should be the infinite series we are trying to prove (aka: 1 + 2 + 3 + 4 + 5......) The first sum is going to be: S1: 1-1+1-1+1-1+1-1+1....There are two possible answers for this infinite sum: If we end after a subtraction, the answer will be 0. If we end after an addition, the answer if 1. So, to compute the sum of the series, we would take the average of the two values ~ (1+0)/2 = 1/2. Step 2. Write the second sum twice.Write the next sum, which is slightly different, involving both addition and subtraction. We are going to duplicate this sum and add it to each other (2S2) to make the work easier. Also, shift the second S2 to one digit right so that the first number of the second sum is below the second number of the first sum. S2: 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8.....+ S2: 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8.....2S2= 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1...When you add both the sums digit by digit, you get the original sum we started with, which is S1. Therefore, we can conclude that 2S2 = S1 = 1/2., which means that S2= 1/4. Step 3. Write the original sum we are solving for and subtract S2 from it.S3: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8.....- [1 - 2 + 3 - 4 + 5 - 6 + 7 - 8.....]0 + 4 + 0 + 8 + 12 + 0 + 16 + 0...Now, we are going to factor this new sum by 4, therefore, we obtain...you got it...the original sum times 4. 4 ( 1 + 2 + 3 + 4 + 5.....) = 4S3S3 - S2 = 4S3 S3 - 1/4 = 4S3 -1/4 = 3S3 -1/12 = S3 And...there is your proof that the sum of the infinite series is -1/12. It is absolutely surprising, but actually proven to be true. I know..weird. But I like it. Good luck,
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Remember these tips for SAT with calculator! These are important:
Since m is the product of all integers: m = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 = 1 x 2 x 3 x (2 x 2) x 5 x (3 x 2) x 7 x (2 x 2 x 2) x (3 x 3) x (5 x 2) <-- there are 8 2's Therefore, the answer to this question is 8 = n. 4. If you are more confident with a sample, there must be a wider range of the numbers, which means a greater margin of error. 5. (IMPORTANT TIP FOR Exponential Growth questions): If they give you an exponential equation for a particular scenario and ask you to find out when it is doubled, do this: For ex. 18 x 16^0.0125t <-- what is the value of t when the equation is doubled? 18 x 2^4 x 0.0125t <- change the 16 to 2^4 18 x 2^0.05t <- simplified <- to find the value of t, simply make 0.05t equal to one so that the equation can double 0.05t = 1 t = 20 Good luck,
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