Infinite alternating series with repeated logarithms

[LeifEricson]

Homework Statement

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.

Homework Equations

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.

 

[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!

 

[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.

 

[LeifEricson]

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

 

[futurebird]

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

 

[Dick]

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

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.