Let $$\sum_{n=0}^{\infty}a_{n}x^{n}$$ a power series with $a_{n}>0,\forall n\geq0$. Calling $R$ its convergence radius, then is true (or not) that the series cannot converge in the interval $$(-R,R]?$$
For me, it can converge, but I really don't know to show it. I don't know if there's something to do about $a_{n}>0$.
We have
$$|a_n(-R)^n|=a_nR^n$$
thus, if $ \sum a_nR^n $ converge, it will be the same for $ \sum a_n(-R)^n$.
the converse is not true.
take $a_n=\frac 1n$.
the radius is $R=1$.
$$\sum a_nR^n=\sum \frac 1n \text{ diverge}$$ and $$\sum a_n(-R)^n=\sum\frac{(-1)^n}{n}\text{ converge}$$