Does the series converge or diverge?

Question

$oo$
$sum_(n=1)1/(n^(1+1/n)$

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Answer 1 · 2018-06-28T09:10:51

It diverges, since it is asymptotically equivalent to $1/n$

Let's use the comparison test. It goes like this: if you want to know if a series $a_n$ converges, and

$\lim_{n\to\infty}\frac{a_n}{b_n} = c$

where both $a_n$ and $b_n$ are sequences with positive terms and $c$ is finite, theny both $a_n$ and $b_n$ converge or diverge.

In this case, since

$\lim_{n\to\infty}\frac{1}{n^{1+\frac{1}{n}}} = \frac{1}{n}$

we may try to use $b_n=1/n$ as comparison. Ideed, we have

$\lim_{n\to\infty}\frac{\frac{1}{n^{1+\frac{1}{n}}}}{1/n} = \lim_{n\to\infty}\frac{n^{-1-1/n}}{n^{-1}}=\lim_{n\to\infty}n^{(-1-1/n)-(-1)}$

$=\lim_{n\to\infty}n^{-1/n} = 1$

So, the two series behave the same. Sine

$\sum_{n=1}^\infty 1/n=\infty$

then your series diverges as well.