ISPA - Integral and Comparison Tests

Integral and Comparison Tests

In the previous lesson the terms of the infinite series were any real numbers, positive, negative, or zero. For purposes of the Integral Test, Comparison Test, and Limit Comparison Test, only terms that are nonnegative numbers will be considered. The rationale for this restriction is that the partial sums of such series form nondecreasing sequences, and nondecreasing sequences that are bounded from above always converge.

Series Terms as Areas of Rectangles and Relationship to Improper Integrals

Think of the terms of a positive series as rectangular areas with each base of length 1 and heights of a₁ , a₂ , a₃ , ... . For example, consider the harmonic series equation image indicator $LaTeX: \sum^{\infty}_{n=1}\frac{1}{n}$ $\sum^{\infty}_{n=1}\frac{1}{n}$ and compare it to its corresponding integral $LaTeX: \int_1^{\infty }\frac{1}{x}dx$ $\int_1^{\infty }\frac{1}{x}dx$ as illustrated in the graph.

integral test comparison to harmonic series

The respective areas represent the total "left-hand" rectangular areas from 1 to equation image indicator . The area under the curve of the function y = 1/x is less than the total area of the rectangles, or written another way, the integral $LaTeX: \int_1^{\infty }\frac{1}{x}dx≤a_1+2_2+a_3+...$ $\int_1^{\infty }\frac{1}{x}dx≤a_1+2_2+a_3+...$ . This improper integral is known to diverge, i.e., the area under the curve is infinite. So the sum of the series equation image indicator $LaTeX: \sum^{\infty}_{n=1}\frac{1}{n}$ $\sum^{\infty}_{n=1}\frac{1}{n}$ must also be infinite and the series is divergent. In general, the series $LaTeX: \sum^{\infty}_{n=1}a_n$ $\sum^{\infty}_{n=1}a_n$ and its related integral $LaTeX: \int_1^{\infty }a(x)dx$ $\int_1^{\infty }a(x)dx$ either both converge or both diverge.

Integral Test

Unlike many of the series previously discussed, it is generally difficult to find an expression for s_n, the nth partial sum, that is simple enough to decide whether or not it possesses a limit as equation image indicator $LaTeX: n \to \infty$ $n \to \infty$ . Determining whether a series is convergent or divergent without explicitly finding its sum can be accomplished using a test involving improper integrals.

Did You Know? An early form on the Integral Test of convergence was developed in India by Madnava in the early 14th century and by his followers at the Kerala School. In Europe, it was later developed by Maclaurin and Cauchy and is sometimes known as the Malaurin-Cauchy Test. The Integral Test states that if f is a continuous, positive, decreasing function on equation image indicator $LaTeX: [1, \infty )\:\text{and}\:a_n=f(n)$ $[1, \infty )\:\text{and}\:a_n=f(n)$ , then the series $LaTeX: \sum^{\infty}_{n=1}a_n$ $\sum^{\infty}_{n=1}a_n$ and the integral $LaTeX: \int_1^{\infty }a(x)dx$ $\int_1^{\infty }a(x)dx$ both converge or both diverge. The next presentation illustrates using the Integral Test on the harmonic series and two other series.

Examples of using the Integral Test to determine convergence or divergence of a series follow.

The Integral Test applied to other series is depicted in the next presentation.

To use an interactive applet to explore how the Integral Test is used to test convergence or divergence, CLICK HERE. Links to an external site.

Convergence and Divergence of the p-Series

The p-series is a generalization of the harmonic series where equation image indicator $LaTeX: \sum^{\infty}_{n=1}\frac{1}{n^P}=\frac{1}{1^P}+\frac{1}{2^P}+\frac{1}{3^P}+...$ $\sum^{\infty}_{n=1}\frac{1}{n^P}=\frac{1}{1^P}+\frac{1}{2^P}+\frac{1}{3^P}+...$ . It would be helpful to know for which values of p the series converges. If p < 0, then $LaTeX: \lim_{n \to \infty }\frac{1}{n^P}=\infty$ $\lim_{n \to \infty }\frac{1}{n^P}=\infty$ . If p = 0, then equation image indicator $LaTeX: \lim_{n \to \infty }\frac{1}{n^P}=1$ $\lim_{n \to \infty }\frac{1}{n^P}=1$ . In either case $LaTeX: \lim_{n \to \infty }\frac{1}{n^P}≠0$ $\lim_{n \to \infty }\frac{1}{n^P}≠0$ , so the given series diverges by the nth Term Test for Divergence. If p > 0, then the function f(x) = 1/x^p is continuous, positive, and decreasing on equation image indicator $LaTeX: [1, \infty )$ $[1, \infty )$ . Recall from the Integration Techniques, L'Hopital's Rule, and Improper Integrals module that $LaTeX: \int_1^{\infty }\frac{dx}{x^P}$ $\int_1^{\infty }\frac{dx}{x^P}$ diverges if p < 1 and converges if p > 1. Then by the Integral Test, the series converges if p > 1 and diverges if 0 < p < 1. (Note that when p = 1, this p-series is the harmonic series.) These conclusions form the basis of the p-Series Test, which states that the p-series equation image indicator $LaTeX: \sum^{\infty}_{n=1}\frac{1}{n^P}$ $\sum^{\infty}_{n=1}\frac{1}{n^P}$ is convergent if p > 1 and divergent if p < 1. Consider the p-series $LaTeX: 1+\frac{1}{\sqrt[3]{2}}+\frac{1}{\sqrt[3]{3}}+...=1+\frac{1}{2^{\frac{1}{3}}}+\frac{1}{3^{\frac{1}{3}}}+...$ $1+\frac{1}{\sqrt[3]{2}}+\frac{1}{\sqrt[3]{3}}+...=1+\frac{1}{2^{\frac{1}{3}}}+\frac{1}{3^{\frac{1}{3}}}+...$ . Since p = 1/3 and equation image indicator $LaTeX: p\le1$ $p\le1$ , then the series diverges by the p-Series Test. The next presentation focuses on using the p-Series Test to determine convergence or divergence.

Comparison Tests

Comparison tests do not require finding an expression for the nth term partial sum. Just as the Integral Test compared an infinite series with a related improper integral, comparison tests compare a given series with a series that is known to be convergent or divergent.

Comparison Tests (Direct Comparison Test)

The Comparison Test is sometimes referred to as the Direct Comparison Test, which is more descriptive of what actually occurs. The Comparison Test states the following:

Suppose that equation image indicator $LaTeX: \sum a_n\:\text{and}\:\sum b_n$ $\sum a_n\:\text{and}\:\sum b_n$ are series with positive terms.

i) If equation image indicator $LaTeX: \sum b_n$ $\sum b_n$ is convergent and a_n < b_n for all n, then $LaTeX: \sum a_n$ $\sum a_n$ is also convergent.

ii) If equation image indicator $LaTeX: \sum b_n$ $\sum b_n$ is divergent and a_n > b_n for all n, then $LaTeX: \sum a_n$ $\sum a_n$ is also divergent.

View the presentation below describing the Comparison Test and illustrating its application to three examples.

To explore the Comparison Test using an interactive applet, CLICK HERE. Links to an external site.

Limit Comparison Test

The next test is particularly useful with series in which a_n is a rational function of n. The Limit Comparison Test states: Suppose equation image indicator $LaTeX: \sum a_n\:\text{and}\:\sum b_n$ $\sum a_n\:\text{and}\:\sum b_n$ are series with positive terms. If $LaTeX: \lim_{n \to \infty }\frac{a_n}{b_n}=c$ $\lim_{n \to \infty }\frac{a_n}{b_n}=c$ , where c is a finite number and c > 0, then either both series converge or both diverge.

View the presentation below illustrating the Limit Comparison Test.

The limit comparison test can be explored using an interactive applet by CLICKING HERE. Links to an external site.

The presentation below includes examples of using both the Limit Comparison Test and the Comparison Test.

Integral and Comparison Tests Practice

PROBLEM: Emily uses the integral test to determine if equation image indicator $LaTeX: \sum^{\infty}_{k=1}\frac{3}{k^2}$ $\sum^{\infty}_{k=1}\frac{3}{k^2}$ converges. She finds that $LaTeX: \int_1^\infty \frac{3}{x^2}dx=3$ $\int_1^\infty \frac{3}{x^2}dx=3$ . She then states that $LaTeX: \sum^{\infty}_{k=1}\frac{3}{k^2}$ $\sum^{\infty}_{k=1}\frac{3}{k^2}$ converges and the sum is 3. What error did she make?

SOLUTION: Emily is correct that the series converges. Her error was saying that the value of the related integral gives the sum of the infinite series. $LaTeX: \sum_{k=1}^{\infty }\frac{3}{k^2}=3+\frac{3}{4}+\frac{3}{9}+...$ $\sum_{k=1}^{\infty }\frac{3}{k^2}=3+\frac{3}{4}+\frac{3}{9}+...$ is greater than 3. text annotation indicator

Integral and Comparison Tests: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of differential equations and general and particular solutions.

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