For which real values of p does the series ∞∑(n=1) n=π*(ln p)^n converge? State reasons for your answer

Question

For which real values of p does the series ∞∑(n=1)    n=π*(ln p)^n converge? State reasons for your answer

anonymous · Answer

This? $$\sum_{j=1}^{\infty} \pi \ln^j(p)?$$

jamesj · Answer

Or this? $$\sum_{n=1}^{\infty} \pi (\ln p)^n$$

anonymous · Answer

Yes.

anonymous · Answer

The second one.

jamesj · Answer

Wel, this is just a geometric series in disguise: a, ar, ar^2, ar^3, ...

When does the sum of a geometric series converge and what are a and r here?

anonymous · Answer

Ah yep.

Converges when r < 1, a = π, r = (lnp)^n  ?

zarkon · Answer

you are a little off there

anonymous · Answer

r=ln(p) and what we both wrote for the series is the exact same. Just different notation.

anonymous · Answer

Since ln(1/e)=-1 and the ln(e)=1 then it has to be:
1/e<p<e

jamesj · Answer

Yes

The infinite sum of a geometric series (ar^j) converges if and only if |r| < 1

anonymous · Answer

Okay awesome. Thanks for all the help everyone.