# Convergence Or Divergence

1. Apr 5, 2013

### whatlifeforme

1. The problem statement, all variables and given/known data
Determine either absolute convergence, conditional convergence or divergence for the series.

2. Relevant equations
$\displaystyle \sum^{∞}_{n=1} \frac{(-1)^n}{5n^{1/4} + 5}$

3. The attempt at a solution
It converges conditionally i know, but i can't figure out how.
1. I applied the alternating series test and concluded that the series converges.
2. I attempted to use the Ratio test for absolute convergence, but l'hopitals is just going in circles. i'm not getting anywhere; the limit n->∞ is ∞/∞.

Last edited: Apr 5, 2013
2. Apr 5, 2013

### Dick

If you mean (-1)^n/(5n^(1/4)+5) try a comparison test with a p-series. The ratio test won't help you.

3. Apr 5, 2013

### Staff: Mentor

Is this what you meant?
$$\sum^{∞}_{n=1} \frac{(-1)^n}{5n^{1/4} + 5}$$

4. Apr 5, 2013

### whatlifeforme

thanks. i fixed it.

5. Apr 6, 2013

### whatlifeforme

how do i prove the p-series? use an integral test? i don't think my instructor will let us simply identify a p-series but wants us to rather prove it.

6. Apr 6, 2013

### Staff: Mentor

A p-series (below) converges or diverges, depending on the value of p.
$$\sum_{n = 1}^{\infty} \frac{1}{n^p}$$
Use the integral test, with f(x) = x-p, and look at cases for p < 1, p = 1, and p > 1. There might already be a proof of this in your book.