Alternating Series

Alternating series

An alternating series is any series whose terms alternate in sign — that is, any series for which the product of any two consecutive terms is negative.

Equivalently, an alternating series is one that can be written in the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=c}^\infty (-1)^{n+d}\, b_n,}

for some fixed integers $c$ and $d$, and sequence of positive terms $b_{n}$. (If the n^th term of the series is called Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n} , as usual, then notice that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n=|a_n|} .)

As with geometric series, we have defined alternating series here in a slightly more general way than is typically done in calculus textbooks. Usually alternating series are either defined quite restrictively as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty (-1)^{n+1} b_n,}

or as being in one of the two forms

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty (-1)^{n} b_n \mbox{ or } \sum_{n=1}^\infty (-1)^{n-1} b_n,}

in all cases the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n} being some sequence of positive terms. It is easy to see that these other definitions are special cases of our formula above.

It should be obvious that we could not hope to write a formula for the partial sums of a general alternating series (besides, of course, the definition of partial sums given earlier), but, perhaps surprisingly, we can say when such a series converges.

An alternating series converges if its terms eventually decrease in magnitude to zero — that is, if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty} |a_n| = 0}

and

the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{|a_n|\}} is eventually decreasing.

However, if either of these conditions are not satisfied, it does not mean that the alternating series must diverge.

Note that any geometric series with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r<0} is alternating. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |r|<1} , then the conditions for convergence of an alternating series will be satisfied. This can be proven in the general case, but we will simply illustrate with an example.

Example

Consider the series

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty \left(-\frac{1}{2}\right)^{n-1}.}

This series can be written in the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty (-1)^{n-1} \left(\tfrac{1}{2}\right)^{n-1},}

and so matches our definition of an alternating series (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=-1} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n=(\tfrac{1}{2})^{n-1}} ).

It is obvious that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty} |a_n| = \lim_{n\to\infty} \left(\frac{1}{2}\right)^{n-1} = 0}

and that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n|=\left(\frac{1}{2}\right)^{n-1}}

is a decreasing sequence, so the series converges (as we knew it must, since it is geometric with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |r|<1} ).

Example

Consider the series

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n}.}

This is an alternating version of the harmonic series. Since

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty} \frac{1}{n} = 0}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n|=\frac{1}{n}}

is a decreasing sequence of terms, the series converges.

Absolute and conditional convergence

At this point we have considered divergent series whose terms have no limit (the arithmetic series) and divergent series whose terms have a limit of zero (the harmonic series). But every convergent series must have terms that converge to zero. So, does this mean convergent series are "all the same"? Definitely not. There are two kinds of convergence that can be thought of as two "strengths" of convergence: absolute and conditional. The distinction is important because there are things you can do with absolutely convergent series that you cannot do with merely conditionally convergent ones. First, though, some definitions:

• A series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \sum_{n=c}^{\infty} a_n} is said to converge absolutely (or to be absolutely convergent) if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \sum_{n=c}^{\infty} |a_n|} converges.
• A series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \sum_{n=c}^{\infty} a_n} is said to converge conditionally (or to be conditionally convergent) if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \sum_{n=c}^{\infty} |a_n|} diverges but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \sum_{n=c}^{\infty} a_n} itself converges.

It should be obvious that this distinction only makes sense for series with a mixture of positive and negative terms. This includes, but is not limited to, alternating series.

Example

We have seen that the harmonic series

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}}

is divergent but its alternating version

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n}{n}}

is convergent. Since the first series may be formed by taking the absolute value of the terms in the second series, we see that the second series is conditionally convergent.

Example

Consider the series

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}.}

This is an alternating series whose terms decrease to zero in magnitude, so it converges. Furthermore, the series formed by the absolute value of the terms

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}}

is a p-series with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=2} , so it converges also. Therefore the original alternating series is absolutely convergent.

Resources

The Alternating Series Test

• The Alternating Series Test Video by James Sousa, Math is Power 4U
• The Alternating Series Test Video by James Sousa, Math is Power 4U
• Apply the Alternating Series Test Video by James Sousa, Math is Power 4U
• Determine if an Alternating Series Converges or Diverges Video by James Sousa, Math is Power 4U
• Determine if an Alternating Series Converges or Diverges Video by James Sousa, Math is Power 4U
• Ex 1: Determine if a Series and an Alternating Series Converge or Diverge Video by James Sousa, Math is Power 4U
• Ex 2: Determine if a Series and an Alternating Series Converge or Diverge Video by James Sousa, Math is Power 4U

• Alternating Series Video by Patrick JMT
• Alternating Series - Basic Example Video by Patrick JMT
• Two Alternating Series Examples Video by Patrick JMT
• More Alternating Series Examples Video by Patrick JMT
• Alternating Series - Another Example 1 Video by Patrick JMT
• Alternating Series - Another Example 2 Video by Patrick JMT
• Alternating Series - Another Example 3 Video by Patrick JMT
• Alternating Series - Another Example 4 Video by Patrick JMT

• The Alternating Series Test Video by Krista King

• The Alternating Series Test Video by The Organic Chemistry Tutor

The Alternating Series Estimation Theorem

• Find the Error in Using a Partial Sum to Approximate the Sum of an Alternating Series Video by James Sousa, Math is Power 4U
• Number of Terms Needed in a Partial Sum to Approximate the Sum of an Alternating Series with Given Error, Video by James Sousa, Math is Power 4U

• Alternating Series Estimation Theorem Video by Patrick JMT
• Alternating Series Error Estimation Video by Patrick JMT

• Alternating Series Estimation Theorem Video by Krista King

• Alternating Series Estimation Theorem Video by The Organic Chemistry Tutor

Absolute Convergence and Conditional Convergence

• Absolutely and Conditionally Convergent Series Video by James Sousa, Math is Power 4U
• Ex 1: Determine if a Series is Absolutely Convergent, Conditionally Convergent, or Divergent Video by James Sousa, Math is Power 4U
• Ex 2: Determine if a Series is Absolutely Convergent, Conditionally Convergent, or Divergent Video by James Sousa, Math is Power 4U
• Ex 3: Determine if a Series is Absolutely Convergent, Conditionally Convergent, or Divergent Video by James Sousa, Math is Power 4U
• Ex 4: Determine if a Series is Absolutely Convergent, Conditionally Convergent, or Divergent Video by James Sousa, Math is Power 4U

• Absolute Convergence, Conditional Convergence, and Divergence Video by Patrick JMT)
• Absolute Convergence, Conditional Convergence - Another Example 1 Video by Patrick JMT
• Absolute Convergence, Conditional Convergence - Another Example 2 Video by Patrick JMT
• Absolute Convergence, Conditional Convergence - Another Example 3 Video by Patrick JMT

• Absolute and Conditional Convergence Video by Krista King

• Absolute Convergence, Conditional Convergence, and Divergence Video by The Organic Chemistry Tutor

Licensing

Content obtained and/or adapted from:

• Series, Wikibooks under a CC BY-SA license