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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