Convergence of $ \sum^{\infty} \frac{\sum_{k=0}^{n} k!}{(n + p)!}$ for $p=2$.

Question

I can't prove the convergence of this serie for $p=1$ and $p=2$. $$ \sum_{n=2}^{\infty}\frac{\sum_{k=0}^{n} k!}{(n + p)!}$$ I have already tried d'Alembert theorem. I don't have any other ideas. Maybe should I find an upper-bound ? (How because I have this sum over the factorial)

Answer 1

• When $p = 1$, we have

  $$ \frac{\sum_{k=0}^{n} k!}{(n+1)!} \geq \frac{n!}{(n+1)!} = \frac{1}{n+1}. $$

  So the series diverges by the comparison test.

• When $p \geq 2$,

  $$ \sum_{k=0}^{n} k! = n! + \sum_{k=0}^{n-1} k! \leq n! + n \cdot (n-1)! = 2\cdot n!. $$

  So the general term is bounded by

  $$ \frac{\sum_{k=0}^{n} k!}{(n+p)!} \leq \frac{2}{(n+1)\cdots(n+p)} \leq \frac{2}{n^p}. $$

  So the series converges by the comparison test.