Integral Test

Calculus 2

Integral Test

Let $\left\{ a_n \right\}_{n=1}^\infty$ be a sequence of positive terms. Suppose there is a positive integer $N$ such that for all $n \geq N$ , $a_n = f(n)$ , where $f(x)$ is a:

Decreasing,
Continuous,
Positive,

function of $x$ . Then, the series $\sum_{n=N}^\infty a_n$ and the integral $\int_N^\infty f(x) \, dx$ both converge or both diverge.

In other words, the Integral Test states that if $f(x)$ is the continuous function corresponding to $a_n$ and $f(x)$ is decreasing, continuous, and positive, then $\sum_{n=N}^\infty a_n$ and $\int_N^\infty f(x) \, dx$ either both converge or both diverge.

Keep in mind the following:

If the series and integral are convergent, they are not necessarily equal.
The integral need only be decreasing eventually on the interval.
The Integral Test can be used to prove the conclusion of the p-Series Test.

Example 1

Determine the convergence/divergence of the series $\sum_{n=1}^\infty \frac{1}{n+2}$ using the Integral Test.

View Answer

Diverges

Example 2

Determine the convergence/divergence of the series $\sum_{n=1}^\infty \frac{8n}{n^2+4}$ using the Integral Test.

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Diverges

Example 3

Determine the convergence/divergence of the series $\sum_{n=1}^\infty \frac{-10}{n^2+1}$ using the Integral Test.

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Inconclusive (terms are not positive)

Example 4

Determine the convergence/divergence of the series $\sum_{n=1}^\infty n e^{-n^2}$ using the Integral Test.

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Converges

Example 5

Show that the p-Series $\sum_{n=1}^\infty \frac{1}{n^p}$ converges for $p > 1$ and diverges for $p \leq 1$ using the Integral Test (assuming $p$ is a positive real constant).

View Answer

Evaluate $\int_1^\infty \frac{1}{x^p} \, dx$ :

For $p > 1$ , the integral converges.
For $p \leq 1$ , the integral diverges.

Thus, the series follows the same behavior as the integral.

Remainder Estimation

For some convergent series, such as geometric series or telescoping series, we can find the sum of the series. For most convergent series, however, we cannot easily find the sum. Nevertheless, we can estimate the sum by adding the first $n$ terms to get the partial sum $S_n$ , but we need to know how far off $S_n$ is from the total sum $S$ .

If a series $\sum a_n$ is known to be convergent by the Integral Test, the remainder $R_n$ measuring the difference between the total sum $S$ and its $n^\text{th}$ partial sum $S_n$ is given by:

R_n = S - S_n = a_{n+1} + a_{n+2} + a_{n+3} + \cdots.

If $\{a_n\}_{n=1}^\infty$ is a sequence of positive terms with $a_k = f(k)$ , where $f(x)$ is a continuous, positive, decreasing function for $x \geq n$ and $\sum_{k=1}^\infty a_k$ converges to $S$ , then the remainder $R_n = |S - S_n|$ satisfies:

\int_{n+1}^\infty f(x) \, dx \leq R_n \leq \int_n^\infty f(x) \, dx.

Example 6

Estimate the sum $S$ of the series $\sum_{n=1}^\infty \frac{1}{n^2}$ with $n = 10$ .

Comparison Tests

Ratio Test