# Another Test for Convergence Question

## Homework Statement

Test the following series for convergence or divergence.
$$\sum_{n=1}^{\infty} \frac{1}{3^{\ln n}}$$

## The Attempt at a Solution

I've tried to compare this to geometric series $3^n$ but obviously the target term is larger overall than its geometric counterpart. Comparing with the reciprocal of $\ln n$ also brings no avail because the term is smaller than the former. Ratio test gives $ρ=1$ which is inconclusive. It certainly passes preliminary test because the limit goes to 0 once n goes to infinity. I haven't done integral test because it's quite ugly and I don't know how to integrate the corresponding function.

Can you guys give me suggestion to tame this series down nicely?

Thank You

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Dick
Homework Helper
Use that 3=e^(ln(3)) and use the laws of exponentials to rearrange it a bit. It becomes a power series.

• 1 person
Edit: Dick beat me to it. Quick.

Last edited:
Office_Shredder
Staff Emeritus
Gold Member
Try using 3 =eln(3) to rewrite your summand

Strange things happened when I arrange the summation.

$3=e^{\ln 3}$
$$3^{\ln n}=(e^{\ln 3})^{\ln n}$$
$$3^{\ln n}=e^{\ln 3 ° \ln n}$$
$$3^{\ln n}=n^{\ln 3}$$

If that's the case then the series must be convergent because of p-test, with p being larger than 1.

Dick
Homework Helper
Strange things happened when I arrange the summation.

$3=e^{\ln 3}$
$$3^{\ln n}=(e^{\ln 3})^{\ln n}$$
$$3^{\ln n}=e^{\ln 3 ° \ln n}$$
$$3^{\ln n}=n^{\ln 3}$$

If that's the case then the series must be convergent because of p-test, with p being larger than 1.
I don't know if I'd call that strange, but it is the correct conclusion.

I don't know if I'd call that strange, but it is the correct conclusion.
Well maybe I'm not quite used to manipulating exponents yet. Thanks for your help guys!