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

1. Oct 5, 2009

### Piamedes

1. The problem statement, all variables and given/known data
Test for convergence the series:
$$a_[n] = \frac{1}{n*n^{\frac{1}{n}}}$$

2. Relevant equations
Various Sequence Convergence Tests

3. 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.

2. Oct 5, 2009

### Billy Bob

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

3. Oct 5, 2009

### Piamedes

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

4. Oct 5, 2009

### Billy Bob

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

5. Oct 5, 2009

### Piamedes

thanks, the comparison with 2 works perfectly