Choosing a Convergence Test

Learning Outcomes

Describe a strategy for testing the convergence of a given series

At this point, we have a long list of convergence tests. However, not all tests can be used for all series. When given a series, we must determine which test is the best to use. Here is a strategy for finding the best test to apply.

Problem-Solving Strategy: Choosing a Convergence Test for a Series

Consider a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex]. In the steps below, we outline a strategy for determining whether the series converges.

Is [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] a familiar series? For example, is it the harmonic series (which diverges) or the alternating harmonic series (which converges)? Is it a [latex]p-\text{series}[/latex] or geometric series? If so, check the power [latex]p[/latex] or the ratio [latex]r[/latex] to determine if the series converges.
Is it an alternating series? Are we interested in absolute convergence or just convergence? If we are just interested in whether the series converges, apply the alternating series test. If we are interested in absolute convergence, proceed to step [latex]3[/latex], considering the series of absolute values [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex].
Is the series similar to a [latex]p-\text{series}[/latex] or geometric series? If so, try the comparison test or limit comparison test.
Do the terms in the series contain a factorial or power? If the terms are powers such that [latex]{a}_{n}={b}_{n}^{n}[/latex], try the root test first. Otherwise, try the ratio test first.
Use the divergence test. If this test does not provide any information, try the integral test.

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Example: Using Convergence Tests

For each of the following series, determine which convergence test is the best to use and explain why. Then determine if the series converges or diverges. If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges.

[latex]\displaystyle\sum _{n=1}^{\infty }\frac{{n}^{2}+2n}{{n}^{3}+3{n}^{2}+1}[/latex]
[latex]\displaystyle\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}\left(3n+1\right)}{n\text{!}}[/latex]
[latex]\displaystyle\sum _{n=1}^{\infty }\frac{{e}^{n}}{{n}^{3}}[/latex]
[latex]\displaystyle\sum _{n=1}^{\infty }\frac{{3}^{n}}{{\left(n+1\right)}^{n}}[/latex]

Show Solution

Step 1. The series is not a [latex]p-\text{series}[/latex] or geometric series.

Step 2. The series is not alternating.

Step 3. For large values of [latex]n[/latex], we approximate the series by the expression

[latex]\frac{{n}^{2}+2n}{{n}^{3}+3{n}^{2}+1}\approx \frac{{n}^{2}}{{n}^{3}}=\frac{1}{n}[/latex].

Therefore, it seems reasonable to apply the comparison test or limit comparison test using the series [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}[/latex]. Using the limit comparison test, we see that

[latex]\underset{n\to \infty }{\text{lim}}\frac{\frac{\left({n}^{2}+2n\right)}{\left({n}^{3}+3{n}^{2}+1\right)}}{\frac{1}{n}}=\underset{n\to \infty }{\text{lim}}\frac{{n}^{3}+2{n}^{2}}{{n}^{3}+3{n}^{2}+1}=1[/latex].

Since the series [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}[/latex] diverges, this series diverges as well.
Step 1.The series is not a familiar series.

Step 2. The series is alternating. Since we are interested in absolute convergence, consider the series

[latex]\displaystyle\sum _{n=1}^{\infty }\frac{3n}{\left(n+1\right)\text{!}}[/latex].

Step 3. The series is not similar to a p-series or geometric series.

Step 4. Since each term contains a factorial, apply the ratio test. We see that

[latex]\underset{n\to \infty }{\text{lim}}\frac{\frac{\left(3\left(n+1\right)\right)}{\left(n+1\right)\text{!}}}{\frac{\left(3n+1\right)}{n\text{!}}}=\underset{n\to \infty }{\text{lim}}\frac{3n+3}{\left(n+1\right)\text{!}}\cdot \frac{n\text{!}}{3n+1}=\underset{n\to \infty }{\text{lim}}\frac{3n+3}{\left(n+1\right)\left(3n+1\right)}=0[/latex].

Therefore, this series converges, and we conclude that the original series converges absolutely, and thus converges.
Step 1. The series is not a familiar series.

Step 2. It is not an alternating series.

Step 3. There is no obvious series with which to compare this series.

Step 4. There is no factorial. There is a power, but it is not an ideal situation for the root test.

Step 5. To apply the divergence test, we calculate that

[latex]\underset{n\to \infty }{\text{lim}}\frac{{e}^{n}}{{n}^{3}}=\infty [/latex].

Therefore, by the divergence test, the series diverges.
Step 1. This series is not a familiar series.

Step 2. It is not an alternating series.

Step 3. There is no obvious series with which to compare this series.

Step 4. Since each term is a power of [latex]n[/latex], we can apply the root test. Since

[latex]\underset{n\to \infty }{\text{lim}}\sqrt[n]{{\left(\frac{3}{n+1}\right)}^{n}}=\underset{n\to \infty }{\text{lim}}\frac{3}{n+1}=0[/latex],

by the root test, we conclude that the series converges.

try it

For the series [latex]\displaystyle\sum _{n=1}^{\infty }\frac{{2}^{n}}{{3}^{n}+n}[/latex], determine which convergence test is the best to use and explain why.

Hint

Show Solution

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In the following table, we summarize the convergence tests and when each can be applied. Note that while the comparison test, limit comparison test, and integral test require the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] to have nonnegative terms, if [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] has negative terms, these tests can be applied to [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex] to test for absolute convergence.

Summary of Convergence Tests
Series or Test	Conclusions	Comments
Divergence Test For any series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex], evaluate [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}[/latex].	If [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}=0[/latex], the test is inconclusive.	This test cannot prove convergence of a series.
	If [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}\ne 0[/latex], the series diverges.	This test cannot prove convergence of a series.
Geometric Series [latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}[/latex]	If [latex]\|r\|<1[/latex], the series converges to [latex]\frac{a}{\left(1-r\right)}[/latex].	Any geometric series can be reindexed to be written in the form [latex]a+ar+a{r}^{2}+\cdots [/latex], where [latex]a[/latex] is the initial term and [latex]r[/latex] is the ratio.
	If [latex]\|r\|\ge 1[/latex], the series diverges.
p-Series [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{p}}[/latex]	If [latex]p>1[/latex], the series converges.	For [latex]p=1[/latex], we have the harmonic series [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}[/latex].
	If [latex]p\le 1[/latex], the series diverges.
Comparison Test For [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with nonnegative terms, compare with a known series [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex].	If [latex]{a}_{n}\le {b}_{n}[/latex] for all [latex]n\ge N[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges.	Typically used for a series similar to a geometric or [latex]p[/latex] -series. It can sometimes be difficult to find an appropriate series.
	If [latex]{a}_{n}\ge {b}_{n}[/latex] for all [latex]n\ge N[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges.
Limit Comparison Test For [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with positive terms, compare with a series [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] by evaluating [latex]L=\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}[/latex].	If [latex]L[/latex] is a real number and [latex]L\ne 0[/latex], then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] both converge or both diverge.	Typically used for a series similar to a geometric or [latex]p[/latex] -series. Often easier to apply than the comparison test.
	If [latex]L=0[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges.
	If [latex]L=\infty [/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges.
Integral Test If there exists a positive, continuous, decreasing function [latex]f[/latex] such that [latex]{a}_{n}=f\left(n\right)[/latex] for all [latex]n\ge N[/latex], evaluate [latex]{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx[/latex].	[latex]{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] both converge or both diverge.	Limited to those series for which the corresponding function [latex]f[/latex] can be easily integrated.
Alternating Series [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}\text{or}\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}[/latex]	If [latex]{b}_{n+1}\le {b}_{n}[/latex] for all [latex]n\ge 1[/latex] and [latex]{b}_{n}\to 0[/latex], then the series converges.	Only applies to alternating series.
Ratio Test For any series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with nonzero terms, let [latex]\rho =\underset{n\to \infty }{\text{lim}}\|\frac{{a}_{n+1}}{{a}_{n}}\|[/latex].	If [latex]0\le \rho <1[/latex], the series converges absolutely.	Often used for series involving factorials or exponentials.
	If [latex]\rho >1\text{or}\rho =\infty [/latex], the series diverges.
	If [latex]\rho =1[/latex], the test is inconclusive.
Root Test For any series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex], let [latex]\rho =\underset{n\to \infty }{\text{lim}}\sqrt[n]{\|{a}_{n}\|}[/latex].	If [latex]0\le \rho <1[/latex], the series converges absolutely.	Often used for series where [latex]\|{a}_{n}\|={b}_{n}^{n}[/latex].
	If [latex]\rho >1\text{or}\rho =\infty [/latex], the series diverges.
	If [latex]\rho =1[/latex], the test is inconclusive.

Activity: Series Converging to [latex]\pi [/latex] and [latex]\frac{1}{\pi} [/latex]

Dozens of series exist that converge to [latex]\pi [/latex] or an algebraic expression containing [latex]\pi [/latex]. Here we look at several examples and compare their rates of convergence. By rate of convergence, we mean the number of terms necessary for a partial sum to be within a certain amount of the actual value. The series representations of [latex]\pi [/latex] in the first two examples can be explained using Maclaurin series, which are discussed in the next chapter. The third example relies on material beyond the scope of this text.

The series

[latex]\pi =4\displaystyle\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{2n - 1}=4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\cdots [/latex]

was discovered by Gregory and Leibniz in the late [latex]1600\text{s}\text{.}[/latex] This result follows from the Maclaurin series for [latex]f\left(x\right)={\tan}^{-1}x[/latex]. We will discuss this series in the next chapter.
1. Prove that this series converges.
2. Evaluate the partial sums [latex]{S}_{n}[/latex] for [latex]n=10,20,50,100[/latex].
3. Use the remainder estimate for alternating series to get a bound on the error [latex]{R}_{n}[/latex].
4. What is the smallest value of [latex]N[/latex] that guarantees [latex]|{R}_{N}|<0.01\text{?}[/latex] Evaluate [latex]{S}_{N}[/latex].
The series

[latex]\begin{array}{cc}\hfill \pi & =6 {\displaystyle\sum _{n=0}^{\infty }} \frac{\left(2n\right)\text{!}}{{2}^{4n+1}{\left(n\text{!}\right)}^{2}\left(2n+1\right)}\hfill \\ & =6\left(\frac{1}{2}+\frac{1}{2\cdot 3}{\left(\frac{1}{2}\right)}^{3}+\frac{1\cdot 3}{2\cdot 4\cdot 5}\cdot {\left(\frac{1}{2}\right)}^{5}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}{\left(\frac{1}{2}\right)}^{7}+\cdots \right)\hfill \end{array}[/latex]

has been attributed to Newton in the late [latex]1600\text{s}\text{.}[/latex] The proof of this result uses the Maclaurin series for [latex]f\left(x\right)={\sin}^{-1}x[/latex].
1. Prove that the series converges.
2. Evaluate the partial sums [latex]{S}_{n}[/latex] for [latex]n=5,10,20[/latex].
3. Compare [latex]{S}_{n}[/latex] to [latex]\pi [/latex] for [latex]n=5,10,20[/latex] and discuss the number of correct decimal places.
The series

[latex]\frac{1}{\pi }=\frac{\sqrt{8}}{9801}\displaystyle\sum _{n=0}^{\infty }\frac{\left(4n\right)\text{!}\left(1103+26390n\right)}{{\left(n\text{!}\right)}^{4}{396}^{4n}}[/latex]

was discovered by Ramanujan in the early [latex]1900\text{s}\text{.}[/latex] William Gosper, Jr., used this series to calculate [latex]\pi [/latex] to an accuracy of more than [latex]17[/latex] million digits in the [latex]\text{mid-}1980\text{s}\text{.}[/latex] At the time, that was a world record. Since that time, this series and others by Ramanujan have led mathematicians to find many other series representations for [latex]\pi [/latex] and [latex]\frac{1}{\pi} [/latex].
1. Prove that this series converges.
2. Evaluate the first term in this series. Compare this number with the value of [latex]\pi [/latex] from a calculating utility. To how many decimal places do these two numbers agree? What if we add the first two terms in the series?
3. Investigate the life of Srinivasa Ramanujan [latex]\left(1887\text{-}1920\right)[/latex] and write a brief summary. Ramanujan is one of the most fascinating stories in the history of mathematics. He was basically self-taught, with no formal training in mathematics, yet he contributed in highly original ways to many advanced areas of mathematics.

Module 5: Sequences and Series