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Investigate sum

by Hydr0matic
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Hydr0matic
#1
May27-04, 07:53 AM
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[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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arildno
#2
May27-04, 08:14 AM
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Quote Quote by Hydr0matic
[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]?
Hydr0matic
#3
May27-04, 08:41 AM
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P: 199
yes.. thnx.

matt grime
#4
May27-04, 08:53 AM
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Investigate sum

Hint: sqrt(k+1)-sqrt(k) = {sqrt(k)+sqrt(k+1)}^{-1}
and you can put the logs together.
Hydr0matic
#5
May27-04, 09:41 AM
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P: 199
Got it. Thnx matt.
yrch
#6
May27-04, 09:46 AM
P: 2
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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