Show convergence of a series involving power series

Question

Suppose that the function $f= f(t)$ is given by a power series

$$f(t) = c_1t + c_2 t^2 + \cdots$$ that converges for all t. Let $a_1, a_2, \cdots$ be such that $\sum_k |a_k| < \infty$. Show that the series $\sum_k f(a_kt)$ converges for all $t$, and that the resulting function is given by an everywhere convergent power series of $t$. Write the coefficients of the power series in terms of the c's and the a's. (Hint: If $|a_k| <M$, then $|a_k^n| \leq |a_k| M^{n-1}$ for $n=1,2,\cdots $)

Attempt: We need to show that $\sum_k f(a_kt)$ converges. This is $\sum_k\sum_j c_j(a_jt)^j$ = $\sum_k\sum_j c_j(a_j)^jt^j$.

Questions: (1) Is this the correct way to approach the problem?

(2) Can I split $\sum_k c_kt^k\sum_j(a_j)^j$? If not, then how do I show convergence for the double sum?

(3) Finally, how to reduce the double sum to a power series?

Answer 1

Let $|a_{k}|<M<1$ for $k\geq N$, then $|a_{k}^{n}|\leq |a_{k}|M^{n-1}$, $n=1,2,...$, and \begin{align*} \sum_{k\geq N}\sum_{n}|c_{n}(a_{k}t)|^{n}&\leq\sum_{k\geq N}\sum_{n}|c_{n}a_{k}|M^{n-1}|t|^{n}\\ &\leq\sum_{k\geq N}\left(\sum_{n}|c_{n}||t|^{n}\right)|a_{k}|\\ &=\left(\sum_{k\geq N}|a_{k}|\right)\left(\sum_{n}|c_{n}||t|^{n}\right)\\ &<\infty. \end{align*} Note that the series $\displaystyle\sum_{k}f(a_{k}t)=\sum_{1\leq k\leq N-1}f(a_{k}t)+\sum_{k\geq N}f(a_{k}t)$, and we have just proved that $\displaystyle\sum_{k\geq N}|f(a_{k}t)|\leq\sum_{k\geq N}\sum_{n}|c_{n}(a_{k}t)^{n}|<\infty$.

Answer 2

Actually, if we only know there is a constant $C$ such that $|f(t)| \le C|t|$ for $|t|\le 1$ (and the given $f$ satisfies this and much more), then

$$\tag 1 \sum_{k=1}^{\infty}|f(a_kt)| < \infty \,\, \text { for all } t \in \mathbb R.$$

Proof: Given $t,$ we can choose $k_0$ such that $k>k_0$ implies $|a_kt| \le 1$ (here using $|a_k| \to 0$). Thus $(1)$ is bounded above by

$$\sum_{k=1}^{k_0}|f(a_kt)| + \sum_{k>k_0}C|a_kt| = \sum_{k=1}^{k_0}|f(a_kt)| + C|t|\sum_{k>k_0}|a_k|<\infty.$$