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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 ?

envy grunt
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2 Answers2

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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.

mathematics2x2life
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  • yes, i think it's correct. But can we also use the Abel's test here, using the fact that $S_n = (1 + \frac{1}{1!} + ... + \frac{1}{n!})$ is bounded by e ? – envy grunt Oct 17 '19 at 20:41
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    @envygrunt Indeed, my test uses Dirichlet's Test but you should be able to use the same sort of breaking up the summand, proving 'nice' properties about the pieces, and proving the sum converges the same way using Abel's Test. Notice the test here requires more on the partial sum, whereas Abel's Test just forces more burden on the sequence. But again, I'm sure you can use the ideas here and do something nearly identical with Abel's Test. A good exercise! – mathematics2x2life Oct 17 '19 at 20:44
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Of course the sequence is bounded,because notice that the sequence is monotonically increasing,and convergent,then it must be bounded.You can use abel's test here.