# Series root or ratio test?

1. Mar 23, 2010

### zeion

1. The problem statement, all variables and given/known data

Determine whether the series converges or diverges.

$$\sum (\sqrt {k} - \sqrt {k - 1})^k$$

2. Relevant equations

3. The attempt at a solution

$$(a_k)^\frac{1}{k} = \sqrt{k} - \sqrt{k - 1}$$

What do I do here..?

$$= \frac{1}{\sqrt{k} + \sqrt{k-1}} \to 0 ?$$
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 23, 2010

### Staff: Mentor

$$\sqrt{k} - \sqrt{k - 1} = \frac{(\sqrt{k} - \sqrt{k - 1})(\sqrt{k} + \sqrt{k - 1})}{\sqrt{k} + \sqrt{k - 1}}$$

3. Mar 23, 2010

### zeion

So that is
$$\sqrt{k} - \sqrt{k - 1} = \frac{(\sqrt{k} - \sqrt{k - 1})(\sqrt{k} + \sqrt{k - 1})}{\sqrt{k} + \sqrt{k - 1}} = \frac {\sqrt{k}^2 - \sqrt{k-1}^2}{\sqrt{k} + \sqrt{k - 1}} = \frac{1}{\sqrt{k} + \sqrt{k-1}}$$

4. Mar 23, 2010

### zeion

Then

$$\frac{1}{\sqrt{k} + \sqrt{k-1}} < \frac{1}{\sqrt{k}} = ( \frac{1}{k})^{\frac{1}{2}$$? then converges?

5. Mar 23, 2010

### Staff: Mentor

You are apparently confusing yourself. You started with the root test, not the comparison test.

BTW, the series whose general term is 1/sqrt(k) diverges, but that's not relevant to what you're doing.

6. Mar 23, 2010

### zeion

Ok so I go here $$\frac{1}{\sqrt{k} + \sqrt{k-1}}$$ and then I'm kinda stuck

7. Mar 23, 2010

### Staff: Mentor

Take the limit as k --> infinity. What do you get? Why is this limit important to you? What did you start out doing in your first post?

8. Mar 23, 2010

### zeion

I get 0? So the ratio is < 1 so its converges?

9. Mar 23, 2010

### Staff: Mentor

The limit is < 1. Try not to confuse yourself into think you are working with the ratio test - here it's the root test, so the fact that you are finding the limit of a fraction is not relevant.

10. Mar 24, 2010

### zeion

Ok so since $$a_k < \mu^k$$ and $$\mu^k$$ converges $$a_k$$converges

11. Mar 24, 2010

### Staff: Mentor

I don't know -- what's $\mu$? That's the first time it has appeared in this thread.

zeion, you need to step back and take a bigger-picture view of what you're doing. You seem to be getting lost in minute details, and losing track of the purpose of the details.