For which values of x does the power series converge or diverge?

Question

First, I know that the series converges when |x+2| < R and diverges when |x+2| > R. So now I have that the radius of convergence is somewhere between 2 and 3. However, this doesn't really give me too good of an indication of the interval of convergence. I would say that since 2 < R < 3 and a=2, then the interval of convergence would be 0 < x < 5, but please correct me on that if I am wrong. I'm kind of just taking a shot in the dark on that one.

So, I'm lead to believe that a and c would be divergent and b and d would be convergent. I just am in need of more insight on this because I'm trying to explain it to someone else but it has been years since I learned the topic so I'm very unsure of my knowledge.

Paul · Answer

The power series must be convergent when $x$ satisfies that $-R <x+2< R$, and be divergent
when $x <-R$ and $x>R$ and be unkown when $x+2=-R$ or $x+2=R$, where $2le R le 3$.

So (a) is convergent; $b$ is divergent; $c$ and $d$ are unknown.

For which values of x does the power series converge or diverge?

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