Identify the alternating signs and magnitudes
Terms that change side
Each bar in the upper graph shows one term of \(2-\frac23+\frac25-\frac27+\cdots\). A green bar lies above the line, so that term is positive, and a red bar lies below the line, so that term is negative.
The orange bars in the lower graph show the size of each term without its sign, so every orange bar lies above the line.
Drag the slider to add terms. The colors alternate, because the sign changes at every step, and the orange bars become shorter.
Each term has two parts: the sign decides the direction of its bar, and the size decides its length.
Any alternating series
A series is alternating when the signs of its terms switch at every step.
Drag the slider to choose a term \(n\). The \(\cg{+}\) or \(\cr{-}\) beside its bar shows its sign, and the orange bar under it shows its size \(a_n\).
Press the button to start with a negative term. Every upper bar turns over, and the orange bars do not move, because the sign and the size are separate facts.
Notation
If the first term is positive, write the \(n\)th term as a sign factor, which is \(+1\) or \(-1\), times a size.
\[(-1)^{n+1}\,\cy{a_n}\]At \(n=1\) the exponent is 2, so the factor is \(+1\). At \(n=2\) the exponent is 3, so the factor is \(-1\).
If the first term is negative, use the sign factor \((-1)^n\) instead. At \(n=1\) this factor is \(-1\).
\[(-1)^{n}\,\cy{a_n}\]The size \(a_n\) is the absolute value of the term, which is its distance from 0, so it is positive for every \(n\).
\[\cy{a_n}=|\text{term}|>0\]To find \(a_n\), remove the sign factor from the formula. What remains must be positive for every \(n\).
When it fails
A series with a sine or a cosine factor often does not alternate. For \(n\) in radians, \(\sin n\) is positive for \(n=1,2,3\) and negative for \(n=4,5,6\).
The upper graph shows three green bars, then three red bars. In an alternating series the color changes at every bar, so \(\sum\frac{2\sin n}{\sqrt n}\) is not alternating.
Example 1: The first term is negative
Identify the sign factor and the size \(a_n\) in \(\displaystyle\sum_{n=1}^{\infty}(-1)^n\frac{5}{n+2}\). Then list the first four terms.
The formula contains the factor \((-1)^n\), which equals \(-1\) at \(n=1\). So the first term is negative and the first bar is red.
\[(-1)^1=-1\]Remove the sign factor. The size is \(a_n=\frac{5}{n+2}\), and it is positive for every \(n\ge1\) because \(n+2\) is positive.
\[\cy{a_n}=\frac{5}{n+2}\]Evaluate the first four terms. The signs switch at every step, so the series is alternating.
\[-\frac53+\frac54-1+\frac56-\cdots\]
Example 2: A sign factor that cancels
Is \(\displaystyle\sum_{n=1}^{\infty}(-1)^n\frac{\cos(n\pi)}{n+1}\) an alternating series?
The formula contains the factor \((-1)^n\), which is \(-1\) for odd \(n\) and \(+1\) for even \(n\). So the series may alternate.
A tempting wrong move: Take \((-1)^n\) as the sign and \(\frac{\cos(n\pi)}{n+1}\) as the size.
\[a_n=\frac{\cos(n\pi)}{n+1}\]A size must be positive, but \(\cos(n\pi)\) is \(-1\) when \(n\) is odd. The middle graph shows red bars there, so this size would be negative for odd \(n\).
Find the values of \(\cos(n\pi)\). It is \(-1\) for odd \(n\) and \(1\) for even \(n\), so \(\cos(n\pi)=(-1)^n\), which is the same sign factor.
\[\cos(n\pi)=(-1)^n\]Multiply the two sign factors. The product is 1, so every term is positive and every bar is green. The series is not alternating.
\[\begin{aligned}(-1)^n(-1)^n&=1\\ \text{term}&=\frac{1}{n+1}\end{aligned}\]
Practice
Write the \(n\)th term of \(\frac32-\frac34+\frac38-\frac3{16}+\cdots\) as a sign factor times a size \(a_n\).
Use the color of the first bar to choose the sign factor. Each orange bar is half as long as the bar before it.
Solution
The first term is positive, so the sign factor is \((-1)^{n+1}\).
\[(-1)^{1+1}=1\]The sizes are \(\frac32\), \(\frac34\), \(\frac38\) and \(\frac3{16}\). The denominator doubles at each step, so it is \(2^n\).
\[\cy{a_n}=\frac{3}{2^n}\]Join the two parts. At \(n=4\) the formula gives \(-\frac3{16}\), which is the fourth term.
\[(-1)^{n+1}\,\frac{3}{2^n}\]
Questions
The graph shows the first six terms of a series. The upper bars are the terms, and the orange bars are their sizes.
What is the size \(a_3\) of the third term?
- 1.2
- −1.2
- 1.5
- 2
Answer
1.2
Which sign factor matches the colors of the upper bars?
- \((-1)^n\)
- \((-1)^{n+1}\)
- \((-1)^{2n}\)
Answer
\((-1)^n\)
Which of these series is alternating?
- \(\sum\frac{\cos(n\pi)}{n+3}\)
- \(\sum(-1)^n\frac{\cos(n\pi)}{n+3}\)
- \(\sum\frac{\cos n}{n+3}\)
- \(\sum\frac{(-1)^{2n}}{n+3}\)
Answer
\(\sum\frac{\cos(n\pi)}{n+3}\)
Summary
You write each term as a sign factor times a positive size \(a_n\), and you confirm that the signs switch at every step.
This works because the sign factor decides the side of the line, and the size decides the distance from the line.
This step comes before the alternating series test, because that test needs alternating signs and then looks only at the sizes \(a_n\).
This is one of four skills in topic 10.7 of Which & Why: AP Calculus.