Investigate sum

1. May 27, 2004

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 ?

2. May 27, 2004

arildno

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

3. May 27, 2004

Hydr0matic

yes.. thnx.

4. May 27, 2004

matt grime

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

5. May 27, 2004

Hydr0matic

Got it. Thnx matt.

6. May 27, 2004

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