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Let $0<a_n<1$, and $a_n\to 1,n\to \infty$.

when $\sum_{n=1}^\infty (a_n)^n$ convergent or divergent?

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2 Answers 2

up vote 7 down vote accepted

If $(a_n)^n=n^{-2}$, then $\sum(a_n)^n$ converges, and $a_n=n^{-2/n}\to1$.

If $(a_n)^n=n^{-1}$, then $\sum(a_n)^n$ diverges, and $a_n=n^{-1/n}\to1$.

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Sounds like homework, so here is a hint: What about $a_n := 1-\frac{1}{n}$? (Use $\frac{1}{e} = \lim_{n \to \infty} (a_n)^n$.)

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