investigate sum

[Hydr0matic]

\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{k+1}-\ln{k})


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

[arildno]

\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{k+1}-\ln{k})


How do I go about finding out if it's convergent or divergent ?
Do you mean:

\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{(k+1)}-\ln{k})
?

[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
\frac{1}{x} \geq \ln (1 + \frac{1}{x})
for all positive x

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