I have mainly two questions, the first one being motivated by the second one.
1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ without using the fact that $F(x) = 1-cos x$?
2) If $(u_k)_{k\in\mathbb{N}}$ is a positive sequence decreasing to $0$, such that $u_k^{1/(2k)}$ converges to $u\in (0,1)$, can I prove that $G(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} u_k\ x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$?
I have tried several things, in particular comparing the sums of these two series, Abel transform, ... Maybe by adding some restriction on $u_k$ (of the form $\frac{u^{2k-2}}{2k}$ would be enough for my purpose)?
Thanks.
Remark: a more general (unanswered) question was raised in: Criteria for boundedness of power series, and the sequence in 2) matches the necesary condition mentioned in this post.