Question

Does the series converge or diverge?

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

Answer 1

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

Explanation:

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.