Is There a Convergence Test for the Series 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