Solving the Mystery of Convergence: \sum_{k=2}^{\infy}(\frac{1}{ln(k!)})

[Emil_]

Homework Statement

I need help to decide if the series below are convergent or divergent.

<br /> \sum_{k=2}^{\infy}(\frac{1}{ln(k!)})<br />

Homework Equations

The Attempt at a Solution

I tried using the d'Alembert ratio test but the ratio is 1 if I calculated it correctly and then nothing can be said about the series.

 

[quantumdude]

Hmmm...I haven't worked it out completely, but you might try a lesser known convergence test called Raabe's test.

http://mathworld.wolfram.com/RaabesTest.html

 

[Emil_]

Thank you. I'll take a look on Raabe's test and see if I can work it out, but I'm pretty sure it could be done by another test since the course I'm taking doesn't teach Raabe's test.

 

[lanedance]

have you heard of stirlings approximation?

 

[Emil_]

have you heard of stirlings approximation?

I think my teacher mentioned briefly an exact formula for n!, involving integrals of arctan etc. He said that the formula was rare even though it was derived 100 years ago, but I've not heard of stirlings approximation. However I think I can solve it using stirlings approximation, thank you!