# investigate sum

by Hydr0matic
Tags: investigate
 P: 199 $$\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{k+1}-\ln{k})$$ How do I go about finding out if it's convergent or divergent ?
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 Quote by 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 ?
Do you mean:
$$\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{(k+1)}-\ln{k})$$?
 P: 199 yes.. thnx.
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P: 9,398

## investigate sum

Hint: sqrt(k+1)-sqrt(k) = {sqrt(k)+sqrt(k+1)}^{-1}
and you can put the logs together.
 P: 199 Got it. Thnx matt.
 P: 2 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

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