Conditions for the convergence of :$\cos\left( \sum_{n\geq0}{a_n}x^n\right)$

Question

let $a_n\geq0 $ be a sequence of the power series defined as : $f(x)=\sum_{n\geq0}{a_n}x^n$ which is convergent for all $x\in \mathbb{R}$, my question here is trivial , under the cited assumption :

Question: Is $\displaystyle \cos\left( \sum_{n\geq0}{a_n}x^n\right)$ convergent ?

Answer 1

The cosine function is continuous hence for all sequences $\left(u_n\right)_{n \in \mathbb{N}}$ such that $u_n \underset{n \rightarrow +\infty}{\rightarrow}u$, $$ \cos\left(u_n\right)\underset{n \rightarrow +\infty}{\rightarrow}\cos\left(u\right) $$ Then, for $x \in \mathbb{R}$ $$ \cos\left(\sum_{n=0}^{+\infty}a_nx^n\right)=\cos\left(f\left(x\right)\right) $$

Answer 2

Set $y:=\sum_{k=0}^\infty a_k x^k$. If $y\in\Bbb R$ then $\cos (y)$ is a number, so in what sense a number is convergent?

Maybe you want to ask if the sequence defined by $\{\cos(y_n)\}_{n\in\Bbb N}$ converges where $y_n:=\sum_{k=0}^na_k x^k$. Then the sequence converges whenever $\{y_n\}_{n\in\Bbb N}$ also converges, because the cosine function is continuous, that is $$\lim_{n\to\infty}\cos(y_n)=\cos\left(\lim_{n\to\infty} y_n\right)=\cos(y)$$