# Convergence of 1/(n*n^(1/n))

Piamedes

## Homework Statement

Test for convergence the series:
$$a_[n] = \frac{1}{n*n^{\frac{1}{n}}}$$

## Homework Equations

Various Sequence Convergence Tests

## The Attempt at a Solution

So far I've tried both a normal comparison and limit comparison test with n^2. The normal one seemed fine until the end. Here was my logic:

For n greater than 1 (its just less than, not equal)

$$n^{\frac{1}{n}} \le n$$

$$\frac{1}{n^{\frac{1}{n}}} \ge \frac{1}{n}$$

$$\frac{1}{n*n^{\frac{1}{n}}} \ge \frac{1}{n^2}}$$

But that doesn't work because it just proves that for every term, this sequence is greater than the p-series for n^2.

For the limit comparison test I don't get an actual limit, so I can't use it.

If anyone has any suggestions for which test to use, or what series to compare it to, I would be most grateful.

Billy Bob
Would $$n^{\frac{1}{n}} \le \log n$$ work?

Piamedes
uh, I'm pretty sure that your inequality is backwards

Billy Bob
uh, I'm pretty sure that your inequality is backwards

I'm pretty sure I'm right for "large" n. And on further reflection, n^(1/n) < 2 is even better.

Piamedes
thanks, the comparison with 2 works perfectly