<s>[INST] Does the series $\sum_{n = 1}^{\infty}\frac{\cos n}{\sqrt{n}}(1 + \frac{1}{1!} + \frac{1}{2!} + ... + \frac{1}{n!})$ converge? I want to check if this series converge or diverge:
$$\sum_{n = 1}^{\infty}\frac{\cos n}{\sqrt{n}}(1 + \frac{1}{1!} + \frac{1}{2!} + ... + \frac{1}{n!})$$
Which technique can i use there ?
UPD: I thought using Abel's theorem here as following. Series 
$$\sum_{n = 1}^{\infty}a_n = \sum_{n = 1}^{\infty} \frac{\cos n }{\sqrt{n}}$$
converges. But is sequence 
$$\{(1 + \frac{1}{1!} + \frac{1}{2!} + ... + \frac{1}{n!})\}_{n = 1}^{\infty}$$
bounded ?
 [/INST] Recall that
$$
e= \lim_{n \to \infty} 1 + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \cdots + \dfrac{1}{n!}
$$
Now consider $S_N= \cos n (1+1/1!+\cdots+1/N!)$. We have $\sum_{n=1}^M S_n$ is bounded for every integer $M$; to see this, consider see this post and use the fact that you know the sum of the reciprocal factorials tends to $e$, and the sequence $\{1/\sqrt{n}\}$ tends to $0$. Therefore, by Dirichlet's Test, $\displaystyle \sum_{n=1}^\infty \dfrac{\cos n}{\sqrt{n}}\left( 1 + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \cdots + \dfrac{1}{n!}\right)$ converges. 
</s>