Integral test for convergence

In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.

Statement of the test

Consider an integer and a function defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series

:$\sum_^\infty f(n)$

converges to a real number if and only if the improper integral

:$\int_N^\infty f(x)\,dx$

is finite. In particular, if the integral diverges, then the series diverges as well.

Remark

If the improper integral is finite, then the proof also gives the lower and upper bounds

for the infinite series.

Note that if the function $f(x)$ is increasing, then the function $-f(x)$ is decreasing and the above theorem applies.

Many textbooks require the function $f$ to be positive, but this condition is not really necessary, since when $f$ is negative and decreasing both $\sum_^\infty f(n)$ and $\int_N^\infty f(x)\,dx$ diverge.

Proof

The proof uses the comparison test, comparing the term $f(n)$ with the integral of $f$ over the intervals continuous almost everywhere">,n+1) respectively.

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