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

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Discussion Overview

The discussion centers around the convergence or divergence of the series \(\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{k+1}-\ln{k})\). Participants explore methods to analyze the series, including manipulation of terms and inequalities.

Discussion Character

  • Exploratory, Mathematical reasoning, Homework-related

Main Points Raised

  • Participants inquire about methods to determine the convergence or divergence of the series.
  • There is clarification on the expression being analyzed, specifically regarding the logarithmic terms.
  • A participant suggests a hint involving the manipulation of the square root term and combining logarithmic expressions.
  • Another participant proposes proving an inequality related to logarithms, indicating a potential approach to the problem.

Areas of Agreement / Disagreement

Participants are engaged in a collaborative exploration of the problem, but no consensus on the convergence or divergence has been reached. Multiple approaches and ideas are presented without resolution.

Contextual Notes

Participants have not fully resolved the mathematical steps involved in analyzing the series, and there may be dependencies on specific definitions or assumptions regarding convergence tests.

Hydr0matic
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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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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]?
 
yes.. thnx.
 
Hint: sqrt(k+1)-sqrt(k) = {sqrt(k)+sqrt(k+1)}^{-1}
and you can put the logs together.
 
Got it. Thnx matt.
 
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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