Suppose we have a real power series $\Sigma a_k X^k$ whose coefficients verify $$\frac{a_{k+1}}{a_k} \to 1$$
It is well known (see d'Alembert ratio test applied to power series) that the radius of convergence of this series is $1$
That is to say, by the definition of the Radius of Convergence, that the series converges inside $(-1,1)$
Question:
Do you agree that this series will not converge at $X=1$ ?
Here is my proof of this:
As $\frac{a_{k+1}}{a_k} \to 1$ there exists an index let's say $k=100$ such that: $$\sum_{k=100} ^{\infty}a_kX^k = a_{100}X^{100}(1+X+X^2+X^3 + ...)$$ $$=a_{100}X^{100}\frac{1}{1-X}$$
which diverges at $X=1$
Now, a subsidiary question would be: can we say that this series will converge at $X=-1$ ? My guess is that it depends...