Is \sum (\sqrt{k+1} - \sqrt{k})(\ln{k+1}-\ln{k}) convergent or divergent?

  • Thread starter Hydr0matic
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In summary, to determine whether the given series is convergent or divergent, you can use the hint provided to rewrite it and then combine the logs. After this, you can use the given inequality to prove that the series is convergent.
  • #1
Hydr0matic
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1
[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 ?
 
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  • #2
Hydr0matic said:
[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 ?
Do you mean:
[tex]
\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{(k+1)}-\ln{k})
[/tex]?
 
  • #3
yes.. thnx.
 
  • #4
Hint: sqrt(k+1)-sqrt(k) = {sqrt(k)+sqrt(k+1)}^{-1}
and you can put the logs together.
 
  • #5
Got it. Thnx matt.
 
  • #6
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
 

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