It's been very long since I studied this but due to COVID i've found myself having to help a younger relative with maths. Today she's presented me with a problem that I have been so far unable to answer, I've been looking through the material that she has from school and haven't been able to find an answer there either. The problem in question looks as follows:
The power series $$\sum_{n=1}^{\infty} c_n x^n$$ converges on the interval $[-2,2)$
a. Evaluate $$\lim_{n\to\infty} c_n$$ b. Evaluate $$\lim_{n\to\infty} nc_n$$
Now the closed interval at -2 irked me but the first thing I thought of doing was trying to find out what $c_n$ was. To do this I used the ratio test and got to a point where I know that $c_n$ needs to be something like $2^{(-n)} * a_n$ where $a_n$ is something that converges when n tends to infinity. However I've been unable to find an $a_n$ that will make the series converge on $x=-2$.
At this point I was quite bothered and thought that maybe there is a simple way to relate the interval of convergence with the limit. Can anyone shed some light into this please?
Since $ 1\in\left[-2,2\right) $, $ \sum\limits_{n\to +\infty}{c_{n}} $ converges, thus $ c_{n}\underset{n\to +\infty}{\longrightarrow}0 \cdot $
The power series $ \sum\limits_{n\geq 0}{n c_{n} x^{n}} $ will also have a radius of convergence $ R=2 $. Hence setting $ x=1 $, we get that$ \sum\limits_{n\geq 0}{n c_{n}} $ converges, which means $ nc_{n}\underset{n\to +\infty}{\longrightarrow}0 \cdot $