Suppose we are given a convergent series $\Sigma a_n$. Then $\lim_{n\rightarrow \infty} a_n = 0$. My question is on whether or not we can conclude that the sequence $\{a_n\}$ is monotonic for all $n > N$ for some given $N \in \mathbb{N}$?
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3No, they may even change signs now and then. Like $a_n=\sin(n)/n^2$. – H. H. Rugh Aug 24 '16 at 21:59
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1Or $\frac{(-1)^n}{n+1}$... or every every term whose index corresponds to a multiple of $7$ is zero (take any converging series, add zeros infinitely often)... basically, no. No monotonicity. – Clement C. Aug 24 '16 at 22:01
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That's a great example. – J. Dunivin Aug 24 '16 at 22:01
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2Every sequence has a monotonic subsequence, but that's about all you can say in general. – Aug 24 '16 at 22:10
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Consider $0 + 1/2 + 0 + 1/2^2 + 0 + 1/2^3 + 0 + \cdots$ – zhw. Aug 24 '16 at 22:15
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Other idea : $(-1/2)^n$ or more generally $q^n$ for $-1<q<0$ which oscillates between positive and negative values and tends to 0. Note that this is the general term of a convergent series $$\sum_{n\geq 0} q^n=\dfrac{1}{1-q}$$
paf
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