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Integral Test

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What is the The Integral Test?

The Integral Test enables us to determine whether a series is convergent or divergent without explicitly finding its sum.

Suppose f is a continuous, positive, decreasing function on and let an = f(n).
Then the series is convergent if and only if the improper integral is convergent.

If is convergent then is convergent.

If is divergent then is divergent.


Test the series for convergence or divergence.


The function is continuous, positive, decreasing function on [1,∞) so we use the Integral Test:

Integral Test

Since is a convergent integral and so, by the Integral test, the series is convergent.

Integral Test - Basic Idea
Using the Integral Test for Series

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