Integral Test

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The Integral Test

Integral Test: If $f$ is a continuous, positive and decreasing function where $f(n)=a_n$ on the interval $[1,\infty)$, then
the improper integral $\displaystyle\int_1^\infty f(x)\, dx$ and the infinite series $\displaystyle\sum_{n=1}^\infty a_n$
either both converge or both diverge.

Picture infinitely many rectangles of width 1 and height $a_n$, so the area of the $n^{th}$ rectangle is $a_n$. Then the series $\displaystyle\sum_{n=1}^\infty a_n$ is equal to the sum of the areas of these infinitely many rectangles. See the graphic examples below.

Consider this graph. We see that the value of the series from $a_2$ on is less than the area under the curve $f$ from 1 to infinity; i.e.
$\displaystyle\sum_{n=2}^\infty a_n<\int_1^\infty f(x)\,dx$. If this integral converges to some finite value $C$, then by using this inequality and doing a little work, we get $\displaystyle\sum_{n=1}^\infty a_n= a_1+\sum_{n=2}^\infty a_n<a_1+\int_1^\infty f(x)\,dx=a_1+C<\infty$, so the series is finite, and thus the series converges.

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Now consider this graph. We see that the sum of the same series beginning with $a_1$ is larger than the area under the same curve $f$ from 1 to infinity; i.e.
$\displaystyle\int_1^\infty f(x)\,dx<\sum_{n=1}^\infty a_n$. If this integral diverges, then because of our constraints on $f$ it diverges to infinity. Since the area under $f$ is infinite, then the sum of the areas of the rectangles must also be infinite, i.e. $\displaystyle\sum_{n=1}^\infty a_n$ is infinite, and thus the series diverges. We see that if the integral diverges, so does the series.

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Summary: either both the integral and the series converge, or both diverge.

From our work with improper integrals, you may have seen that the improper integral

$\displaystyle\int_1^\infty\frac{1}{x^p}\,dx$ converges if $p>1$, and diverges if $p\le 1$.

By using the integral test, we therefore get our $p$-series test, which is extremely useful, especially when used to find comparable series for the comparison tests.

$\displaystyle{\sum_{n=1}^\infty \frac{1}{n^p}}$ converges if $p>1$ and diverges if $p \le 1$.

Explanation and examples of the integral test, as well as determining the above integral of $\frac{1}{x^p}$ and the $p$-series test are included on the first video. The second video includes detail of the graphical information above.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

The Integral Test