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

## 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 $$\sum$$ an diverges, while the improper integral from 1 to infinity of f converges. Justify your answer.

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## 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.

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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.

This reasoning applies only to decreasing functions.
Try something which is "large" for integers, but vanishes outside of small intervals around them.

This is Dr. Block, please remove this question.