absolute convergence

[cheffy]

\(\displaystyle \
\sum\limits_{n = 0}^\infty {\left( {x + 5} \right)^n }
\\)

I need to find the interval of convergence, the radius of convergence, and where it converges absolutely and where it converges conditionally.

I think the interval is -6<x<-4 and the radius is 1, but I'm not sure and I'm not sure how to solve the rest.

Thanks!

 

[soroban]

Hello, cheffy!

\(\displaystyle \L\sum_{n = 0}^\infty (x\,+\,5)^n\)

Find the interval of convergence, the radius of convergence,
and where it converges absolutely and where it converges conditionally.

I think the interval is -6 < x < -4 and the radius is 1. .Yes!

Ratio test: \(\displaystyle \L\:R \:=\: \left|\frac{a_{n+1}}{a_n}\right|\:=\:\left|\frac{(x\,+\,5)^{n+1}}{x\,+\,5)^n}\right| \:=\:|x\,+\,5| \:<\:1\)

Then: \(\displaystyle \:-1 \:< \:x\,+\,5\:<\:1\;\;\Rightarrow\;\;\L\fbox{-6\:<\:x\:<\:-4}\;\) Interval of convergence

. . Radius of convergence: \(\displaystyle \L\fbox{1}\)


Let \(\displaystyle r\:=\:x\,+\,5\)
If \(\displaystyle x\) is in the interval, we have: \(\displaystyle \L\:\sum_{n=0}^{\infty} r^n\;\)where \(\displaystyle |r|\,<\,1\)

. . which is a geometric series which converges absolutely.


At the endpoints, we have: \(\displaystyle \L\:\sum_{n=0}^{\infty}1\,\) and \(\displaystyle \L\,\sum_{n=0}^{\infty}(-1)^n\)

. . Both series diverge.