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Give an example of a series that diverges, but where the improper integral converges.

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  • #1
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Homework Statement



Give an example of a continuous, non-negative function f: [1, infinity) --> R such that if an = f(n) for each positive integer n, the series [tex]\sum[/tex] an diverges, while the improper integral from 1 to infinity of f converges. Justify your answer.

Homework Equations


N/A


The Attempt at a Solution


I have tried to randomly pick some functions I thought would work but can't seem to get one to converge and the other to diverge. For instance the series (1/n) will obviously diverge (p-test) but the integral of (1/n) will be the ln(n) which would give me an infinite value when taking the limit as n approaches infinity. Any advice on what type of function would work.
 

Answers and Replies

  • #2
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I don't think such function exists.

Proof: Assume such function f(n) exists. Since integral from 1 to infinity converges, the integral test tells us that the sum from 1 to infinity also converges. This contradicts the fact we want the sum to diverge. Therefore no such function exists.
 
  • #3
133
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This reasoning applies only to decreasing functions.
Try something which is "large" for integers, but vanishes outside of small intervals around them.
 
  • #4
3
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This is Dr. Block, please remove this question.
 

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