Find the nature of $\sum_{n = 1}^\infty \frac{1! + 2! + \cdots + n!}{(n + 2)!}$

Question

I need to find whether the following series converges or diverges: $$\sum_{n = 1}^\infty \frac{1! + 2! + \cdots + n!}{(n + 2)!}$$

I've plotted the graph of this series using Desmos and it seems to converge. So I tried finding a series with bigger terms that converges but I couldn't find one… Can you help me, please?

Answer 1

Hint: Note that $$ 1 \le \frac{1}{{n!}}\sum\limits_{k = 1}^n {k!} = \frac{1}{{n!}}\left( {n! + \sum\limits_{k = 1}^{n - 1} {k!} } \right) \le \frac{1}{{n!}}\left( {n! + (n - 1)(n - 1)!} \right) \le \frac{1}{{n!}}(n! + n!) = 2 $$ for all $n\geq 1$.

Answer 2

We have that $\sum_{k=1}^{n} k! \sim n!$ and therefore

$$\frac{1! + 2! + \cdots + n!}{(n + 2)!} \sim \frac{n!}{(n+2)!}=\frac{1}{(n+2)(n+1)}$$

therefore the given series converges by limit comparison test.

Refer to the related

• How to approximate $\sum_{k=1}^n k!$ using Stirling's formula?