Limit of a Sequence 05

Calculus Level pending

Let a n a_n be a real-valued sequence such that a n > 0 \displaystyle{a_n > 0} for all n ≥ N n \geq N , for some N ∈ N N \in \mathbb{N} , and ∑ n = 1 ∞ a n \displaystyle{\sum_{n=1}^{\infty}a_n} converges to a finite value. Which of the following is the STRONGEST true statement that can be made about a n \displaystyle{a_n} ?

lim ⁡ n → ∞ a n \lim_{n \rightarrow \infty} a_n exists and is equal to 0 0 . None of these are necessarily true. lim ⁡ n → ∞ a n \lim_{n \rightarrow \infty} a_n exists and is some finite value. lim ⁡ n → ∞ a n \lim_{n \rightarrow \infty} a_n does not exist.

Daniel Juncos by Daniel Juncos , 45, USA

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1 solution

Daniel Juncos
Sep 17, 2017

If a n a_n does not converge to 0 0 , then there is some ϵ > 0 \displaystyle{\epsilon > 0} and some subsequence a n k \displaystyle{a_{n_k}} of a n a_n such that a n k ≥ ϵ \displaystyle{a_{n_k} \geq \epsilon} for all n k n_k . Then ∑ n = 1 ∞ a n ≥ ∑ k = 1 ∞ a n k ≥ ∑ k = 1 ∞ ϵ = ϵ ∑ k = 1 ∞ 1 = ∞ \displaystyle{\sum_{n=1}^{\infty}a_n \geq \sum_{k=1}^{\infty}a_{n_k} \geq \sum_{k=1}^{\infty}\epsilon = \epsilon \sum_{k=1}^{\infty}1 = \infty} .

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