Integral Test

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

The Integral Test

Suppose f is a continuous, positive, decreasing function on and let a_n = 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.

Example:

Test the series for convergence or divergence.

Solution:

The function is continuous, positive, decreasing function on so we use the Integral Test:

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

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