Question

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

Answer 1

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

Answer 2

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

Answer 3

Yes.

Answer 4

The second one.

Answer 5

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?

Answer 6

Ah yep.

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

Answer 7

you are a little off there

Answer 8

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

Answer 9

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

Answer 10

Yes

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

Answer 11

Okay awesome. Thanks for all the help everyone.