How to test this serie for convergence?

[tamtam402]

Homework Statement

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

Homework Equations

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.

 

[micromass]

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

 

[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.

 

[DryRun]

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

That an ingenious way of solving this problem.

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

 

[tamtam402]

y = 3^ln(x)

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

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

I'm stuck here

 

[Bohrok]

3^{ln x} = e^{ln(3ln x)} = e^{(ln x)(ln 3)} = (e^{ln x})^{ln 3} = x^{ln 3} (and here we have a nice log/exponential identity!)

So 1/3^{ln n} = 1/n^{ln 3}

 

[tamtam402]

Thanks, that's very clever!