anonymous 3 years ago Do the following series converge?

1. anonymous

$\sum_{n=1}^{\infty} \frac{ e^{-\sqrt{n}} }{ \sqrt{n} }$ $\sum_{n=2}^{\infty} \frac{ 1 }{ n(\ln n)^3 }$

2. geerky42

I don't do this thing yet so I may be wrong... Ask yourself a question. Which increase faster as n approaches to infinity? $$\large e^{-\sqrt{n}}$$ or $$\sqrt{n}$$. If $$\large e^{-\sqrt{n}}$$ increase faster or approach to something greater than $$\sqrt{n}$$, then it diverges. Otherwise it converges. What does $$n(\ln n )^3$$ approach to when n approach to infinity? If it approaches to something under one, then it diverges, otherwise it converges.

3. amoodarya

for the first use ratio test that is converegent and for the second use integral test see what i draw|dw:1360527817600:dw| that is also conv

4. anonymous

Thank you!