(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Is the series [tex]\Sigma[/tex][tex]\stackrel{\infty}{k=1}[/tex] ([tex]\sqrt{k+1}[/tex] - [tex]\sqrt{k}[/tex])/k convergent or divergent?

2. Relevant equations

The Comparison Test:

0<=a_{k}<=b_{k}

1.The series [tex]\Sigma[/tex][tex]\stackrel{\infty}{k=1}[/tex] a_{k}converges if the series [tex]\Sigma[/tex][tex]\stackrel{\infty}{k=1}[/tex] b_{k}converges.

2. The series [tex]\Sigma[/tex][tex]\stackrel{\infty}{k=1}[/tex] b_{k}diverges if the series [tex]\Sigma[/tex][tex]\stackrel{\infty}{k=1}[/tex] a_{k}diverges.

3. The attempt at a solution

I computed the equation until it looked like this: [tex]\Sigma[/tex][tex]\stackrel{\infty}{k=1}[/tex] ([tex]\sqrt{1/k + 1/k^2}[/tex] - 1/[tex]\sqrt{k}[/tex]) and then I tried to find some other series that would be smaller than the original but still diverge because my guess is that this series diverges. But if I take the series [tex]\Sigma[/tex][tex]\stackrel{\infty}{k=1}[/tex] ([tex]\sqrt{1/k^2}[/tex] - 1/[tex]\sqrt{k}[/tex]) the terms of the series become negative and the rules of the Comparison Test don't apply.

Then I tried the series [tex]\Sigma[/tex][tex]\stackrel{\infty}{k=1}[/tex] ([tex]\sqrt{k+1}[/tex] - [tex]\sqrt{k-1}[/tex])/k and computed it to this: [tex]\Sigma[/tex][tex]\stackrel{\infty}{k=1}[/tex] ([tex]\sqrt{1/k + 1/k^2}[/tex] - [tex]\sqrt{1/k-1/k^2}[/tex]) but it's even harder to analyse than the original series.

Any hints? I also tried series that are greater than the original but found all of them divergent so they weren't of any help.

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# Homework Help: The divergence of a series

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