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Show that if $\sum_{n=1}^{\infty}{a_n}$ converges, then $\lim_{n \rightarrow \infty}{na_n}=0$

Suppose $\lim_{n \rightarrow \infty}{na_n} \neq0$.

Since $\sum_{n=1}^{\infty}{a_n}$ converges, by divergence test , we have $\lim_{n \rightarrow \infty}{a_n}=0$. Then we have for large $N$, $|a_n| < \epsilon$

Since $\lim_{n \rightarrow \infty}{na_n} \neq0$, there exists a large $N_1$ such that $|na_n-L|< \epsilon$. Then I get stuck at here. Can anyone guide me ?

amWhy
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Idonknow
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    Idonknow: You're on the right track...But I think this has been asked here a number of times, and I know for a fact that it was recently asked. I'll search for a link, but don't be surprised/alarmed if it's closed as a duplicate. – amWhy Apr 27 '13 at 15:56
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    You forgot assumptions: $\displaystyle \sum_n\frac{(-1)^n}n$ converges. – Andrés E. Caicedo Apr 27 '13 at 15:59
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    But in my previous, we know that the sequence is decreasing, while in this question we don know that – Idonknow Apr 27 '13 at 16:23
  • Idonknow: If there are no other assumptions, besides the given convergent series, then it's not a true claim: see the counterexample provided by Andres. – amWhy Apr 27 '13 at 16:33

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