8.7 Summary of Tests

OBJECTIVES

When you finish this section, you should be able to:

  1. Choose an appropriate test to determine whether a series converges (p. 596)

In the previous sections, we have discussed a variety of tests that can be used to determine if a series converges or diverges. In the exercises following each section, the instructions indicate which test to use. Unfortunately, in practice, series do not come with instructions. This section summarizes the tests that we have discussed and gives some clues as to what test has the best chance of answering the fundamental question, ‘Does the series converge or diverge?’

1 Choose an Appropriate Test to Determine Whether a Series Converges

The following outline is a guide to help you choose a test to use when determining the convergence or divergence of a series. Table 5 lists the tests we have discussed. Table 6 describes important series we have analyzed.

597

Guide to Choosing a Test to Determine Whether a Series Is Convergent

  1. Check to see if the series is a geometric series or a \(p\)-series. If yes, then use the conclusion given for these series in Table 6.
  2. Find \(\lim\limits_{n\,\rightarrow \,\infty }a_{n}\) of the series \(\sum\limits_{k\,=\,1}^{\infty }a_{k}.\) If \(\lim\limits_{n\,\rightarrow \,\infty }\,a_{n}\neq 0,\) then by the Test for Divergence, the series diverges.
  3. If the series \(\sum\limits_{k\,=\,1}^{\infty }a_{k}\) has only positive terms and meets the conditions of the Integral Test, find the related function \(f.\) Use the integral test if \(\int_{1}^{\infty } f(x)\,dx\) is easy to find.
  4. If the series \(\sum\limits_{k\,=\,1}^{\infty }a_{k}\) has only positive terms and the \(n\)th term is a quotient of sums or differences of powers of \(n\), the Limit Comparison Test with an appropriate \(p\)-series will usually work.
  5. If the series \(\sum\limits_{k\,=\,1}^{\infty }a_{k}\) has only positive terms and the preceding attempts fail, then try the Comparison Test for Convergence or the Comparison Test for Divergence.
  6. Series with some negative terms.
    • For an alternating series, use the Alternating Series Test. It is sometimes better to use the Absolute Convergence Test first.
    • For other series containing negative terms, always use the Absolute Convergence Test first.
  7. If the series \(\sum\limits_{k\,=\,1}^{\infty }a_{k}\) has nonzero terms that involve products, factorials, or powers, the Ratio Test is a good choice.
  8. If the series \(\sum\limits_{k\,=\,1}^{\infty }a_{k}\) has nonzero terms and the \(n\)th term involves an \(n\)th power, try the Root Test.
Table 5: TABLE 5 Tests for Convergence and Divergence of Series
Test Name Description Comment
Test for Divergence for all series (p. 566) \(\sum\limits_{k=1}^{\infty }\,a_{k}\) diverges if \(\lim\limits_{n\,\rightarrow \,\infty}a_{n}\neq 0\). No information is obtained about convergence if \(\lim\limits_{n\,\rightarrow \,\infty}a_{n}=0\).
Integral Test (p. 569) for series of positive terms \(\sum\limits_{k=1}^{\infty }\,a_{k}\) converges (diverges) if \(\int_{1}^{\infty}\,f(x)\,dx\) converges (diverges), where \(f\) is continuous, positive, and decreasing for \(x\geq 1;\) and \(f(k)=a_{k}\) for all \(k\). Good to use if \(f\) is easy to integrate.
Comparison Test for Convergence for series of positive terms (p. 576) \(\sum\limits_{k=1}^{\infty }\,a_{k}\) converges if \(0< a_{k}\leq b_{k}\) and the series \(\sum\limits_{k=1}^{\infty }b_{k}\) converges. \(\sum\limits_{k=1}^{\infty }b_{k}\) must have positive terms and be convergent.
Comparison Test for Divergence for series of positive terms (p. 576) \(\sum\limits_{k=1}^{\infty}\,a_{k}\) diverges if \(a_{k}\geq c_{k}> 0\) and the series \(\sum\limits_{k=1}^{\infty }c_{k}\) diverges. \(\sum\limits_{k=1}^{\infty }c_{k}\) must have positive terms and be divergent.
Limit Comparison Test (p. 577) for series of positive terms \(\sum\limits_{k=1}^{\infty}\,a_{k}\) converges (diverges) if \(\sum\limits_{k=1}^{\infty }b_{k}\) converges (diverges), and \(\lim\limits_{n\,\rightarrow \,\infty }\dfrac{a_{n}}{b_{n}}=L,\) a positive real number. \(\sum\limits_{k=1}^{\infty}b_{k}\) must have positive terms, whose convergence (divergence) can be determined.
Alternating Series Test (p. 582) \(\sum\limits_{k=1}^{\infty}(-1)^{k+1}a_{k}\), \(a_{k}>0,\) converges if
  • \( \lim\limits_{n\rightarrow \infty }a_{n}=0\) and
  • the \(a_{k}\) are nonincreasing.
The error made by using the \(n\)th partial sum as an approximation to the sum \(S\) of the series is less than the \((n+1) \)st term of the series.
Absolute Convergence Test (p. 586) If \(\sum\limits_{k=1}^{\infty}\,\left\vert a_{k}\right\vert \) converges, then \(\sum\limits_{k=1}^{\infty}a_{k}\) converges. The converse is not true. That is, if \(\sum\limits_{k=1}^{\infty}\,\left\vert a_{k}\right\vert \) diverges, \(\sum\limits_{k=1}^{\infty }a_{k}\) may converge.
Ratio Test (p. 591) for series with nonzero terms \(\sum\limits_{k=1}^{\infty }a_{k}\) converges if \(\lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert <1\).
\(\sum\limits_{k=1}^{\infty}a_{k}\) diverges if \(\lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert >1\) or
if \(\lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert = \infty\).		 Good to use if \(a_{n}\) includes factorials or powers.
It provides no information if \(\lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert =1.\)
Root Test (p. 593) for series with nonzero terms \(\sum\limits_{k=1}^{\infty }a_{k}\) converges if \(\lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\vert a_{n}\vert }<1\).
\( \sum\limits_{k=1}^{\infty }a_{k}\) diverges if \(\lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\vert a_{n}\vert }>1\) or
if \(\lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\vert a_{n}\vert }=\infty\).		 Good to use if \(a_{n}\) involves \(n\)th powers.
It provides no information if \(\lim\limits_{n\,\rightarrow\,\infty }\sqrt[n]{\vert a_{n}\vert }=1\).

598

Table 6: TABLE 6 Important Series
Series Name Series Description Comments
Geometric series (p. 557) \(\sum\limits_{k=1}^{\infty }ar^{k-1}=a+ar+ar^{2} +\cdots ,~a\neq 0\) Converges to \(\dfrac{a}{1-r}\) if \(\vert r\vert <1\); diverges if \(\vert r\vert \geq 1.\)
Harmonic series (p. 561) \(\sum\limits_{k=1}^{\infty }\dfrac{1}{k}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots \) Diverges.
p-series (p. 570) \(\sum\limits_{k=1}^{\infty }\dfrac{1}{k^{p}}=1+\dfrac{1}{2^{p}}+\dfrac{1}{3^{p}}+\cdots \) Converges if \(p>1\); diverges if \(0<p\leq 1\).
\(k\)-to-the-\(k\) series (p. 576) \(\sum\limits_{k=1}^{\infty }\dfrac{1}{k^{k}}=1+\dfrac{1}{2^{2}}+\dfrac{1}{3^{3}}+\dfrac{1}{4^{4}}+\cdots \) Converges.
Factorial series (p. 591) \(\sum\limits_{k\,=\,0}^{\infty }\dfrac{1}{k!}=1+1+\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{24}+\cdots \) Converges.
Alternating harmonic series (p. 583) \(\sum\limits_{k=1}^{\infty }\dfrac{(-1)^{k+1}}{k}=1-\dfrac{1}{2}+\dfrac{1}{3}- \dfrac{1}{4}+\cdots \) Converges.