# Alternating Series and P-series convergence

1. Jul 28, 2008

### kylera

Alternating Series and P-series "convergence"

I couldn't resist trying out a pun. Anyway, onto the question:

1. The problem statement, all variables and given/known data
Test the series for convergence/divergence:
$$\sum^{\infty}_{n=1}\frac{-1^{n-1}}{\sqrt{n}}$$

2. Relevant equations
Alternating Series Test and possibly p-series test...

3. The attempt at a solution
The expression for an alternating series goes as a(n) = (-1)^(n-1) * b(n). Having said that, it's obvious that b(n) is

$$\frac{1}{\sqrt{n}}$$

which can be re-written as $$\frac{1}{n^\frac{1}{2}}$$. But by the rules of the p-series, since 0.5 is obviously lesser than 1, I came to the conclusion that the series diverges. Instead, the answer states that the series does indeed converge. Can anyone help shed some light in this?

2. Jul 28, 2008

### rock.freak667

Re: Alternating Series and P-series "convergence"

I think for the p-series test, the numerator must be '1' only.

3. Jul 28, 2008

### Dick

Re: Alternating Series and P-series "convergence"

Forget the p-series test. Concentrate on the alternating series test. b(n) is decreasing towards 0. Sorry, I don't get the 'pun'. Am I being thick?

4. Jul 28, 2008

### d_leet

Re: Alternating Series and P-series "convergence"

I don't think you are since I don't get it either.

5. Jul 28, 2008

### rock.freak667

Re: Alternating Series and P-series "convergence"

Well usually for the p-series test, the only examples with fractions I've ever seen is where the function is in the form

$$\frac{1}{f(n)}$$

6. Jul 28, 2008

### Dick

Re: Alternating Series and P-series "convergence"

Sure, in fact, it applies only to series of the form 1/n^p or n^(p). I wasn't saying you were wrong. It's useful here if you want to discuss absolute convergence. It is useless for the given function.

7. Jul 29, 2008

### HallsofIvy

Staff Emeritus
Re: Alternating Series and P-series "convergence"

The "p-series" only applies to positive series. That's why Dick says it is useful to discuss absolute convergence. If your series has both positive and negative terms then it may converge "conditionally". It will converge absolutely only if the series of absolute values converges.

By the "alternating series test", the series
$$\sum \frac{(-1)^n}{\sqrt{n}}$$
converges.

By the "p- series test"
$$\sum \left|\frac{(-1)^n}{\sqrt{n}}\right|= \sum \frac{1}{\sqrt{n}}= \sum \frac{1}{n^{1/2}}$$
does NOT converge and so the original series converges conditionally, not absolutely.