Section 4.1 Summary of Series
In previous sections, we explored various methods for determining if an infinite series converges or diverges. Also, in a few limited cases, it is possible to determine the exact value of the convergent series.
The culmination of the series chapter is to be able to take a given series and determine if it converges or diverges. Most problems won’t tell you which test to use, so you’ll have to be able to use pattern recognition to figure out which test applies. This is similar to how evaluating integrals involves pattern recognition for which technique to use. For each of these questions, you should:
Subsection 4.1.1 Choosing Which Test to Use
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Do the terms not approach 0?If \(\lim_{n\to\infty}a_n\neq0\) or the limit does not exist, use the \(n\)th-term test for divergence: the series diverges immediately.
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Are there exponents of \(n\text{?}\) Could be a geometric series. Rewrite in the form \(\sum a\,r^n\text{.}\) With common ratio \(r\text{,}\)The sum is \(\frac{\text{first term}}{1-r}\) (if it converges)
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Are two similar terms being subtracted? Or can you do partial-fraction decomposition? Could be telescoping. Write the partial sum \(s_n\) explicitly, then take \(\lim_{n \to \infty} s_n\text{.}\) Often used for simple rational functions, or with quadratic denominators that are factorable.
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Can it be written as a power of \(n\text{?}\)Consider the \(p\)-series \(\sum \frac{1}{n^p}\text{.}\) Use the \(p\)-series test: converges if \(p\gt 1\text{,}\) diverges if \(p \leq 1\text{.}\) Should be second-nature in order to help with the comparison tests.
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Is there a fraction, with a numerator or denominator with more than one term? Especially a rational function, or generally terms involving fractions. Consider comparison tests. Use asymptotic comparison (keeping only the dominant term). Use the direct or limit comparison test: compare to a simpler series (usually a \(p\)-series or geometric).
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Use direct comparison, if it’s easy enough.
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Use limit comparison if direct comparison is too difficult. Calculate \(L = \lim_{n\to\infty}\frac{a_n}{b_n}\text{.}\) If \(L\gt 0\text{,}\) then \(\sum a_n\) and \(\sum b_n\) behave the same.
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Has factorials (\(n!\)) and/or exponentials (\(b^n\))? Especially mixed with powers (\(n^c\)) or double exponentials (\(n^n\)).Use the ratio test: compute \(\rho = \lim_{n\to\infty}\frac{a_{n+1}}{a_n}\text{.}\)
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\(\rho\lt 1\Rightarrow\) converges (absolutely)
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\(\rho\gt 1\Rightarrow\) diverges
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\(\rho=1\Rightarrow\) inconclusive (try another test)
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Has a double power? With \(n\) in the exponent and \(n\) in the base (like \(n^n\) or of the form \((f(n))^n\)).Use the root test: compute \(L = \lim_{n\to\infty}\sqrt[n]{\abs{a_n}}\text{.}\)
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\(L\lt 1\Rightarrow\) converges (absolutely)
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\(L\gt 1\Rightarrow\) diverges
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\(L=1\Rightarrow\) inconclusive (try another test)
Note: Don’t confuse this with a geometric series \(r^n\text{,}\) which has a constant base. -
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Use the alternating series test: if \(\lim_{n\to\infty}a_n=0\) and \(a_n\) is decreasing, then the series converges (conditionally).
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Can be integrated easily, maybe using \(u\)-substitution? Use the integral test: convergence of \(\sum a_n\) is the same as \(\int f(x) \,dx\text{.}\)
Subsection 4.1.2 Overall Remarks
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Some series can be shown to converge or diverge using multiple different tests (which give you the same conclusion). However, one method may be easier or harder than the other.
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Always check if the terms approach 0. The divergence test is the easiest to apply (because it only involves taking a limit), and it can lead to a quick conclusion of divergence. It is often forgotten by students.
