I am looking at these two sums:
$$s_N=\sum_{n=0}^N\frac{\sin n}{n!}x^n, \qquad \text{and} \qquad c_N=\sum_{n=0}^N\frac{\cos n}{n!}x^n, \quad \text{for} \quad x\in \mathbb{R}. $$
I am interested in the the expression $Q(x,N)=\sqrt{s_N^2+c_N^2}$ and the question of whether it converges towards a function or not. In particular, I have the suspicion that:
$$\lim_{N \to \infty}Q(x,N) = e^{qx}, \qquad \text{where} \qquad 0.5403 < q < 0.5404.$$
I only suspect this because I plugged $Q$ into a graphics calculator and nudged $q$ around for a bit. But is my assumption true? How can I prove that $Q$ diverges or converges?
When trying to calculate $s_N^2+c_N^2$, I tried the formula
$$\left( \sum_{i=0}^N a_i \right)^2= \sum_{i=0}^N a_i^2+2\sum_{i<j}^N a_ia_j$$
which, by using $\sin^2n+\cos^2n =1,$ gives me
$$s_N^2+c_N^2 = \sum_{n=0}^N\left( \frac{x^{2n}}{n!^2} \right) + 2\sum_{n<m}^N \frac{\sin(n)\sin(m)+\cos(m)\cos(n)}{n!m!}x^{n+m},$$
but I don't know how to continue from here.