Prove $f_n=\sum_{k=0}^{\infty}a_k(1-\frac{1}{n})^k$ converges to $\infty$.

Question

Given that $\;\sum a_k$ is a divergent series in $(0,\infty)$ and $\sum a_kX^k$ has radius of convergence $\rho_a=1$. This is an exercise from Amann Herbert Analysis.

It gives a hint that we can use the Bernoulli inequality to get an upper bound for terms of the form $1-(1-\frac{1}{n})^k$.

Answer 1

Let $M \in (0,\infty)$. Choose $N$ such that $ \sum\limits_{k=1}^{N} a_k >2M$. Observe that $ \sum\limits_{k=1}^{N} a_k [1-(1-\frac 1 n)^{k}] <\frac 1 2\sum\limits_{k=1}^{N} a_k $ for $n$ sufficiently large. Hence $ \sum\limits_{k=1}^{N} a_k (1-\frac 1 n)^{k}>\frac 1 2\sum\limits_{k=1}^{N} a_k >M$. It follows that $ \sum\limits_{k=1}^{\infty} a_k (1-\frac 1 n)^{k}>M$ for $n$ sufficiently large.