Question

Do the following series converge?

Answer 1

\[\sum_{n=1}^{\infty} \frac{ e^{-\sqrt{n}} }{ \sqrt{n} }\]

\[\sum_{n=2}^{\infty} \frac{ 1 }{ n(\ln n)^3 }\]

Answer 2

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.

Answer 3

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