Question

Constant series quick question :D

Answer 1

\[\sum_{n=1}^{\infty} \frac{ (-1)^{n-1} }{ n7^n }\] Why is this only convergent and not absolute convergent?

Answer 2

I used the Absolute Convergence test \[\sum_{n=1}^{\infty}|a _{n}|\] to then use the Ratio Test which ends up being convergent but somehow my notes only specify as it being convergent and not Absolute Convergent. I want to know if there is a reason as to why in this case it's only convergent and not Absolute Convergent.

Answer 3

I'd find it hard to believe that \(\frac{1}{n7^n}\) doesn't converge.

Answer 4

First we have: \[
n>1\implies \frac{1}{n7^n}<\frac{1}{7^n}
\]

Answer 5

Then we have: \[
2^n<7^n \implies \frac{1}{7^n} < \frac{1}{2^n}
\]

Answer 6

Book might be wrong

Answer 7

I mean I know it converges but what I'm wondering is if this Absolute converge and why not?

Answer 8

it should be absolutely convergent

Answer 9

Alright, just making sure since I only left it as convergent on my notes and I usually explain and solve it until the end. Thanks