Suppose $\sum_{k=0}^{\infty} {a_k}{x^k}$ is a power series and $$\lim_{k\rightarrow\infty}|a_k|^\dfrac{1}{k}{=L>0}$$ converges.
In part a) I used properties of exponential and logarithms to find that
$\quad k^{1/k}\leq (k+1)^{1/k}\leq (2k)^{1/k} \quad$ for every $k\in \mathbb{N}$
part b) asked to used the squeeze lemma to prove that $$\lim_{k\rightarrow\infty}(k+1)^{1/k}={1}$$
part c) asked to prove R=1/L This was done using the root test.
Now I'm stuck on part d) which asks to Prove that $\sum_{k=0}^{\infty} \dfrac{a_k}{k+1}{r^{(k+1)}}$
convergences for $r\in $ (-R,R).
I'm not sure how to proceed. Any help would be greatly appreciated. Thank you,
\begin{align*} \dfrac{|a_{k}|^{1/k}|r|^{(k+1)/k}}{(k+1)^{1/k}}=\dfrac{|a_{k}|^{1/k}|r|^{1/k}r}{(k+1)^{1/k}}\rightarrow\dfrac{L\cdot 1\cdot |r|}{1}=L\cdot|r|<1, \end{align*} if $|r|<R$.