investigate sum

Hydr0matic

[tex]
\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{k+1}-\ln{k})
[/tex]

How do I go about finding out if it's convergent or divergent ?

arildno

Do you mean:
[tex]
\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{(k+1)}-\ln{k})
[/tex]?

Hydr0matic

yes.. thnx.

matt grime

Hint: sqrt(k+1)-sqrt(k) = {sqrt(k)+sqrt(k+1)}^{-1}
and you can put the logs together.

Hydr0matic

Got it. Thnx matt.

yrch

after combining the logs, try to prove that
[tex]\frac{1}{x} \geq \ln (1 + \frac{1}{x})[/tex]
for all positive x

edit:
oops i missed the last reply while typing mine sorry