# Infinite alternating series with repeated logarithms

1. Apr 8, 2009

### LeifEricson

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

Calculate the sum of the following series:
$$\sum_{i=2}^{\infty}(-1)^i \cdot \lg ^{(i)} n$$

Where (i) as a super-script signifies number of times lg was operated i.e. $$\lg ^{(3)} n = (\lg (\lg (\lg n)))$$, and n is a natural number.

2. Relevant equations

3. The attempt at a solution

I proved by Leibniz test that this series converges. But I don't know how to find the number to which it converges to.

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

2. Relevant equations

3. The attempt at a solution

2. Apr 8, 2009

### futurebird

I really don't know how to help you... but since you know it converges you can use the numerical estimate to see if it looks like any famous number?

I hope someone helps I'm really interested in this!

3. Apr 8, 2009

### Dick

If you try doing a numerical estimate it will blow up. Which is informative. Eventually ln^(k)(n)<1. Then ln^(k+1)(n)<0. Now what's ln^(k+2)(n)?? The series isn't even well defined much less convergent.

4. Apr 8, 2009

### LeifEricson

You are right.
The series isn't well defined so I cancel this question. I apologize.

5. Apr 8, 2009

### futurebird

Oh that makes me feel a little better. I could not even see how you had it converging.

:tongue:

6. Apr 8, 2009

### Dick

No need for apologies!! It's a perfectly valid question. It makes a good 'think about it' exercise. It does look at first glance like it ought to be a good candidate for an alternating series test.

Last edited: Apr 8, 2009