I'm having a hard time with this homework question. I can see that the inequalities hold and each of the three term converge to 1. But I'm not sure what what they mean by writing out the definition of each term. Any help would be greatly appreciated thanks.
Here is the question:
Suppose $\quad\displaystyle \sum^\infty_{k=0}a_kx^k\quad$ is a power series and $\quad \lim\limits_{k\rightarrow\infty} |a^k|^{1/k}=L>0 > \quad$ converges.
- Prove that $\quad k^{1/k}\leq (k+1)^{1/k}\leq (2k)^{1/k} \quad$ for every $k\in \mathbb{N}$ by writing out the definition of each term
and using that log and exp are increasing functions.