# How to test this serie for convergence?

1. May 19, 2012

### tamtam402

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

I'm trying to determine if Ʃ 1/(3^ln(n)) converges.

2. Relevant equations

3. The attempt at a solution

The preliminary test isn't of any help since lim n→∞ an = 0.

I tried the integral test but I couldn't integrate the function, and I don't think it's the best way to proceed. I couldn't do anything with the ratio test either, since I don't know how to simplify the 1/(3^ln(n+1)) term.

2. May 19, 2012

### micromass

Staff Emeritus
Do you know what $e^{ln(x)}$ is??

3. May 19, 2012

### tamtam402

I know it equals x.

So the sum of (3^ln(x))^-1 is smaller than the sum of (x)^-1. I guess I could use a comparison test here, thanks.

4. May 19, 2012

### sharks

That an ingenious way of solving this problem.

I might have solved it using micromass' hint. Try letting: $y = 3^{\ln n}$

5. May 19, 2012

### tamtam402

y = 3^ln(x)

ln(y) = ln[3^ln(x)]

ln(y) = ln(x) ln(3)

I'm stuck here

6. May 19, 2012

### Bohrok

3ln x = eln(3ln x) = e(ln x)(ln 3) = (eln x)ln 3 = xln 3 (and here we have a nice log/exponential identity!)

So 1/3ln n = 1/nln 3

7. May 20, 2012

### tamtam402

Thanks, that's very clever!