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)

 

[tacman]

you can give me.**

The first series, {\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {\frac{1}{k}} } \right)^{ - 1} } \right]} }, is known as the harmonic series and it is a well-known example of a divergent series. This means that the series does not have a finite sum and it continues to increase without bound as more terms are added.

On the other hand, the second series, {\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {k!} } \right)^{ - 1} } \right]} }, is known as the factorial series and it is also a divergent series. This series increases at a much faster rate than the harmonic series and it also does not have a finite sum.

In summary, neither of these series converge. They both continue to increase without bound and do not have a finite sum.