Do These Mathematical Series Converge?

[bomba923]

(This isn't homework )

Does this series converge?
$${\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {\frac{1}{k}} } \right)^{ - 1} } \right]} }$$

Does this series converge?
$${\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {k!} } \right)^{ - 1} } \right]} }$$

**I would appreciate any help

 

[matt grime]

The second one obviously converges, and almost as obviously the first diverges. (I hope...)

Bound the internal sum above by something and below by something in the first and second respectively, so that after inverting you're bounding each term in the big sum below and above. If done correctly you see that the first diverges (hint: think harmonic) and the second converges (think exponential)