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I don't get it. How do we exactly know that the sequence is decreasing and $lim a_n=0$?

J.doe
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  • The problem uses non-standard terminology. However, if the series $\sum a_n$ converges, where the $a_k$ are $\ge 0$, then $a_n\to 0$. In particular after a while $0\le a_n\le 1$ and therefore $0\le a_n^p\le a_n$. – André Nicolas Feb 20 '16 at 22:54

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It doesn't need to be decreasing. I suppose that what is meant is "convergent". Just the fact that $\sum a_n$ is convergent implies that $a_n \to 0$ because if $(S_n)$ denotes the sequence of partial sums, we have $a_n = S_n - S_{n-1}$, hence $\lim a_n = \lim(S_n - S_{n-1}) = 0$.