1. ## Infinite Series (15).

Hello
two hard problems:
1) Find a series $\displaystyle \sum a_n$ involves factorial but the ratio test can not determine its convergence, i.e. $\displaystyle \lim_{n\to\infty} \frac{a_{n+1}}{a_n}=1$
2) Use only the comparison test to show that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n!}$ converges provided that the series (which you will use in the comparison) satisfy the conditions of the integral test and its nth term is integrable.

Actually I tried many times.
but they are too hard to me

and this is not for my homework
Actually, we do not have homeworks in my college
and sorry for my bad english

2. For 2

1/n! < 1/n^2 n> 4

3. How about $\displaystyle \sum_{n=1}^{\infty} \frac{n!}{n!+1}$ for the first.

4. Originally Posted by Danny
How about $\displaystyle \sum_{n=1}^{\infty} \frac{n!}{n!+1}$ for the first.
Thanks
But it must have "one" factorial

5. but you got the idea!

6. Originally Posted by Krizalid
but you got the idea!
No. It didn't.
I still can not catch it.

7. Any one ??