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Alternating Series

Use the alternating series test to test an alternating series for convergence.

The Alternating Series Test

A series whose terms alternate between positive and negative values is an alternating series. For example, the series

\[\sum _{n=1}^{\infty }{(-\frac{1}{2})}^{n}=-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\text{\cdots }\]

and

\[\sum _{n=1}^{\infty }\frac{{(-1)}^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\text{\cdots }\]

are both alternating series.

Series (1), shown in , is a geometric series. Since \(|r|=|\text{-}1\text{/}2|<1,\) the series converges. Series (2), shown in , is called the alternating harmonic series. We will show that whereas the harmonic series diverges, the alternating harmonic series converges.

To prove this, we look at the sequence of partial sums \(\{{S}_{k}\}\) ().

Condensed — the full section is in OpenStax Calculus Volume 2.

Remainder of an Alternating Series

It is difficult to explicitly calculate the sum of most alternating series, so typically the sum is approximated by using a partial sum. When doing so, we are interested in the amount of error in our approximation. Consider an alternating series

\[\sum _{n=1}^{\infty }{(-1)}^{n+1}{b}_{n}\]

satisfying the hypotheses of the alternating series test. Let \(S\) denote the sum of this series and \(\{{S}_{k}\}\) be the corresponding sequence of partial sums. From , we see that for any integer \(N\ge 1,\) the remainder \({R}_{N}\) satisfies

\[|{R}_{N}|=|S-{S}_{N}|\le |{S}_{N+1}-{S}_{N}|={b}_{n+1}.\]

In other words, if the conditions of the alternating series test apply, then the error in approximating the infinite series by the \(N\text{th}\) partial sum \({S}_{N}\) is in magnitude at most the size of the next term \({b}_{N+1}.\)

Example

Try it.

Consider the alternating series

\[\sum _{n=1}^{\infty }\frac{{(-1)}^{n+1}}{{n}^{2}}.\]

Use the remainder estimate to determine a bound on the error \({R}_{10}\) if we approximate the sum of the series by the partial sum \({S}_{10}.\)

Solution

From the theorem stated above,

\(|{R}_{10}|\le {b}_{11}=\frac{1}{{11}^{2}}\approx 0.008265.\)

Absolute and Conditional Convergence

Consider a series \(\sum _{n=1}^{\infty }{a}_{n}\) and the related series \(\sum _{n=1}^{\infty }|{a}_{n}|.\) Here we discuss possibilities for the relationship between the convergence of these two series. For example, consider the alternating harmonic series \(\sum _{n=1}^{\infty }{(-1)}^{n+1}\text{/}n.\) The series whose terms are the absolute value of these terms is the harmonic series, since \(\sum _{n=1}^{\infty }|{(-1)}^{n+1}\text{/}n|=\sum _{n=1}^{\infty }1\text{/}n.\) Since the alternating harmonic series converges, but the harmonic series diverges, we say the alternating harmonic series exhibits conditional convergence.

By comparison, consider the series \(\sum _{n=1}^{\infty }{(-1)}^{n+1}\text{/}{n}^{2}.\) The series whose terms are the absolute values of the terms of this series is the series \(\sum _{n=1}^{\infty }1\text{/}{n}^{2}.\) Since both of these series converge, we say the series \(\sum _{n=1}^{\infty }{(-1)}^{n+1}\text{/}{n}^{2}\) exhibits absolute convergence.

As shown by the alternating harmonic series, a series \(\sum _{n=1}^{\infty }{a}_{n}\) may converge, but \(\sum _{n=1}^{\infty }|{a}_{n}|\) may diverge. In the following theorem, however, we show that if \(\sum _{n=1}^{\infty }|{a}_{n}|\) converges, then \(\sum _{n=1}^{\infty }{a}_{n}\) converges.

Condensed — the full section is in OpenStax Calculus Volume 2.

Alternating Series

State whether each of the following series converges absolutely, conditionally, or not at all.

In each of the following problems, use the estimate \(|{R}_{N}|\le {b}_{N+1}\) to find a value of \(N\) that guarantees that the sum of the first \(N\) terms of the alternating series \(\sum _{n=1}^{\infty }{(-1)}^{n+1}{b}_{n}\) differs from the infinite sum by at most the given error. Calculate the partial sum \({S}_{N}\) for this \(N.\)

For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false.

The following series do not satisfy the hypotheses of the alternating series test as stated.

In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.

The following alternating series converge to given multiples of \(\pi .\) Find the value of \(N\) predicted by the remainder estimate such that the \(N\text{th}\) partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(N\) for which the error bound holds, and give the desired approximate value in each case. Up to \(15\) decimals places, \(\pi =3.141592653589793\text{\ldots }.\)

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

For each of the following alternating series, determine whether the series converges or diverges.
1. \(\sum _{n=1}^{\infty }{(-1)}^{n+1}\text{/}{n}^{2}\)
2. \(\sum _{n=1}^{\infty }{(-1)}^{n+1}n\text{/}(n+1)\)
Atskleisti atsakymą
1. Since
  \(\frac{1}{{(n+1)}^{2}}<\frac{1}{{n}^{2}}\ \text{and}\ \frac{1}{{n}^{2}}\to 0,\)
  the series converges.
2. Since \(n\text{/}(n+1)↛0\) as \(n\to \infty ,\) we cannot apply the alternating series test. Instead, we use the nth term test for divergence. Since
  \(\underset{n\to \infty }{\text{lim}}\frac{{(-1)}^{n+1}n}{n+1}\ne 0,\)
  the series diverges.
Determine whether the series \(\sum _{n=1}^{\infty }{(-1)}^{n+1}n\text{/}{2}^{n}\) converges or diverges.

Atskleisti atsakymą

The series converges.
Consider the alternating series
\[\sum _{n=1}^{\infty }\frac{{(-1)}^{n+1}}{{n}^{2}}.\]
Use the remainder estimate to determine a bound on the error \({R}_{10}\) if we approximate the sum of the series by the partial sum \({S}_{10}.\)

Atskleisti atsakymą

From the theorem stated above,
\(|{R}_{10}|\le {b}_{11}=\frac{1}{{11}^{2}}\approx 0.008265.\)
Find a bound for \({R}_{20}\) when approximating \(\sum _{n=1}^{\infty }{(-1)}^{n+1}\text{/}n\) by \({S}_{20}.\)

Atskleisti atsakymą

\(0.04762\)
For each of the following series, determine whether the series converges absolutely, converges conditionally, or diverges.
1. \(\sum _{n=1}^{\infty }{(-1)}^{n+1}\text{/}(3n+1)\)
2. \(\sum _{n=1}^{\infty }\text{cos}(n)\text{/}{n}^{2}\)
Atskleisti atsakymą
1. We can see that
  \[\sum _{n=1}^{\infty }|\frac{{(-1)}^{n+1}}{3n+1}|=\sum _{n=1}^{\infty }\frac{1}{3n+1}\]
  diverges by using the limit comparison test with the harmonic series.
  \[\underset{n\to \infty }{\text{lim}}\frac{1\text{/}(3n+1)}{1\text{/}n}=\frac{1}{3}\]
  Thus, applying Theorem 5.13, the series cannot converge absolutely. Moreover, because of the alternating series test, we can see that the series converges.
  \[\frac{1}{3(n+1)+1}<\frac{1}{3n+1}\ \text{and}\ \frac{1}{3n+1}\to 0,\]
  We can conclude that \(\sum _{n=1}^{\infty }{(-1)}^{n+1}\text{/}(3n+1)\) converges conditionally.
2. Noting that \(|\text{cos}\ n|\le 1,\) to determine whether the series converges absolutely, compare
  \[\sum _{n=1}^{\infty }|\frac{\text{cos}\ n}{{n}^{2}}|\]
  with the series \(\sum _{n=1}^{\infty }1\text{/}{n}^{2}.\) Since \(\sum _{n=1}^{\infty }1\text{/}{n}^{2}\) converges, by the comparison test, \(\sum _{n=1}^{\infty }|\text{cos}\ n\text{/}{n}^{2}|\) converges, and therefore \(\sum _{n=1}^{\infty }\text{cos}\ n\text{/}{n}^{2}\) converges absolutely.
Determine whether the series \(\sum _{n=1}^{\infty }{(-1)}^{n+1}n\text{/}(2{n}^{3}+1)\) converges absolutely, converges conditionally, or diverges.

Atskleisti atsakymą

The series converges absolutely.
Use the fact that
\[1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\text{\cdots }=\text{ln}\ 2\]
to rearrange the terms in the alternating harmonic series so the sum of the rearranged series is \(3\ \text{ln}(2)\text{/}2.\)

Atskleisti atsakymą

Let
\[\sum _{n=1}^{\infty }{a}_{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\frac{1}{8}+\text{\cdots }.\]
Since \(\sum _{n=1}^{\infty }{a}_{n}=\text{ln}(2),\) by the algebraic properties of convergent series,
\[\sum _{n=1}^{\infty }\frac{1}{2}{a}_{n}=\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\text{\cdots }=\frac{1}{2}\sum _{n=1}^{\infty }{a}_{n}=\frac{\text{ln}\ 2}{2}.\]
Now introduce the series \(\sum _{n=1}^{\infty }{b}_{n}\) such that for all \(n\ge 1,\) \({b}_{2n-1}=0\) and \({b}_{2n}={a}_{n}\text{/}2.\) Then
\[\sum _{n=1}^{\infty }{b}_{n}=0+\frac{1}{2}+0-\frac{1}{4}+0+\frac{1}{6}+0-\frac{1}{8}+\text{\cdots }=\frac{\text{ln}\ 2}{2}.\]
Then using the algebraic limit properties of convergent series, since \(\sum _{n=1}^{\infty }{a}_{n}\) and \(\sum _{n=1}^{\infty }{b}_{n}\) converge, the series \(\sum _{n=1}^{\infty }({a}_{n}+{b}_{n})\) converges and
\[\sum _{n=1}^{\infty }({a}_{n}+{b}_{n})=\sum _{n=1}^{\infty }{a}_{n}+\sum _{n=1}^{\infty }{b}_{n}=\text{ln}\ 2+\frac{\text{ln}\ 2}{2}=\frac{3\ \text{ln}\ 2}{2}.\]
Now adding the corresponding terms, \({a}_{n}\) and \({b}_{n},\) we see that
\[\begin{array}{ll}\sum _{n=1}^{\infty }({a}_{n}+{b}_{n}) & =(1+0)+(-\frac{1}{2}+\frac{1}{2})+(\frac{1}{3}+0)+(-\frac{1}{4}-\frac{1}{4})+(\frac{1}{5}+0)+(-\frac{1}{6}+\frac{1}{6}) \\ & \ \\ & \ +(\frac{1}{7}+0)+(-\frac{1}{8}-\frac{1}{8})+\text{\cdots } \\ & =1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\text{\cdots }.\end{array}\]
We notice that the series on the right side of the equal sign is a rearrangement of the alternating harmonic series. Since \(\sum _{n=1}^{\infty }({a}_{n}+{b}_{n})=3\ \text{ln}(2)\text{/}2,\) we conclude that
\[1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\text{\cdots }=\frac{3\ \text{ln}(2)}{2}.\]
Therefore, we have found a rearrangement of the alternating harmonic series having the desired property.
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\frac{n}{n+3}\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\frac{\sqrt{n}+1}{\sqrt{n}+3}\)

Atskleisti atsakymą

Does not converge by divergence test. Terms do not tend to zero.
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\frac{1}{\sqrt{n+3}}\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\frac{\sqrt{n+3}}{n}\)

Atskleisti atsakymą

Converges conditionally by alternating series test, since \(\sqrt{n+3}\text{/}n\) is decreasing. Does not converge absolutely by comparison with p-series, \(p=1\text{/}2.\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\frac{1}{n\text{!}}\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\frac{{3}^{n}}{n\text{!}}\)

Atskleisti atsakymą

Converges absolutely by limit comparison to \({3}^{n}\text{/}{4}^{n},\) for example.
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}{(\frac{n-1}{n})}^{n}\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}{(\frac{n+1}{n})}^{n}\)

Atskleisti atsakymą

Diverges by divergence test since \(\underset{n\to \infty }{\text{lim}}|{a}_{n}|=e.\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}{\text{sin}}^{2}n\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}{\text{cos}}^{2}n\)

Atskleisti atsakymą

Does not converge. Terms do not tend to zero.
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}{\text{sin}}^{2}(1\text{/}n)\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}{\text{cos}}^{2}(1\text{/}n)\)

Atskleisti atsakymą

\(\underset{n\to \infty }{\text{lim}}{\text{cos}}^{2}(1\text{/}n)=1.\) Diverges by divergence test.
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\text{ln}(1\text{/}n)\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\text{ln}(1+\frac{1}{n})\)

Atskleisti atsakymą

Converges by alternating series test.
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\frac{{n}^{2}}{1+{n}^{4}}\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}\frac{{n}^{e}}{1+{n}^{\pi }}\)

Atskleisti atsakymą

Converges conditionally by alternating series test. Does not converge absolutely by limit comparison with p-series, \(p=\pi -e\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}{2}^{1\text{/}n}\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}{n}^{1\text{/}n}\)

Atskleisti atsakymą

Diverges; terms do not tend to zero.
\(\sum _{n=1}^{\infty }{(-1)}^{n}(1-{n}^{1\text{/}n})\) (Hint: \({n}^{1\text{/}n}\approx 1+\text{ln}(n)\text{/}n\) for large \(n.)\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}n(1-\text{cos}(\frac{1}{n}))\) (Hint: \(\text{cos}(1\text{/}n)\approx 1-1\text{/}{n}^{2}\) for large \(n.)\)

Atskleisti atsakymą

Converges by alternating series test. Does not converge absolutely by limit comparison with harmonic series.
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}(\sqrt{n+1}-\sqrt{n})\) (Hint: Rationalize the numerator.)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}})\) (Hint: Find common denominator then rationalize numerator.)

Atskleisti atsakymą

Converges absolutely by limit comparison with p-series, \(p=3\text{/}2,\) after applying the hint.
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}(\text{ln}(n+1)-\text{ln}\ n)\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}n({\text{tan}}^{-1}(n+1)-{\text{tan}}^{-1}n)\) (Hint: Use Mean Value Theorem.)

Atskleisti atsakymą

Converges by alternating series test since \(n({\text{tan}}^{-1}(n+1)\text{-}{\text{tan}}^{-1}n)\) is decreasing to zero for large \(n.\) Does not converge absolutely by limit comparison with harmonic series after applying hint.
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}({(n+1)}^{2}-{n}^{2})\)
\(\sum _{n=1}^{\infty }{(-1)}^{n+1}(\frac{1}{n}-\frac{1}{n+1})\)

Atskleisti atsakymą

Converges absolutely, since \({a}_{n}=\frac{1}{n}-\frac{1}{n+1}\) are terms of a telescoping series.
\(\sum _{n=1}^{\infty }\frac{\text{cos}(n\pi )}{n}\)
\(\sum _{n=1}^{\infty }\frac{\text{cos}(n\pi )}{{n}^{1\text{/}n}}\)

Atskleisti atsakymą

Terms do not tend to zero. Series diverges by divergence test.
\(\sum _{n=1}^{\infty }\frac{1}{n}\ \text{sin}\ (\frac{n\pi }{2})\)
\(\sum _{n=1}^{\infty }\text{sin}(n\pi \text{/}2)\text{sin}(1\text{/}n)\)

Atskleisti atsakymą

Converges by alternating series test. Does not converge absolutely by limit comparison with harmonic series.
[T] \({b}_{n}=1\text{/}n,\) error \(<{10}^{-5}\)
[T] \({b}_{n}=1\text{/}\text{ln}(n),\) \(n\ge 2,\) error \(<{10}^{-1}\)

Atskleisti atsakymą

\(\text{ln}(N+1)>10,\) \(N+1>{e}^{10},\) \(N\ge 22026;\) \({S}_{22026}=-0.9743\text{\ldots }\)
[T] \({b}_{n}=1\text{/}\sqrt{n},\) error \(<{10}^{-3}\)

Symbols used here

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\approx

approximately equal
Equal to the precision shown, not exactly.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

Euler's number
2.71828…, the base whose exponential is its own derivative.

\sin,\ \cos,\ \tan

sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\lim_{x \to a} f(x)

limit
The value f(x) approaches as x approaches a.

f'(x),\ \frac{dy}{dx}

derivative
Instantaneous rate of change; slope of the graph.

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

y',\ y''

first and second derivative of y
Prime notation for derivatives with respect to x (or t).

C,\ C_1,\ C_2

arbitrary constants
Constants of integration fixed by initial conditions.

How to: Alternating Series

Use the alternating series test to test an alternating series for convergence.
Estimate the sum of an alternating series.
Explain the meaning of absolute convergence and conditional convergence.
Since
Since
We can see that
Noting that
For an alternating series

Questions people ask

What is a derivative in one sentence?

The slope of the graph at a point — the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate — the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function — a polynomial times an exponential, log or trig function.

Pabandyk savo pačių

Parts of this page are adapted from OpenStax Calculus Volume 2 (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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Limits Derivatives Integrals Definite integrals Taylor series Series and sums Maxima and minima The chain rule Implicit differentiation Related rates and optimisation Integration techniques: substitution, parts, partial fractions Applications of integration: area, volume, arc length Infinite series and convergence tests