Unit 10 · Topic 10.10

The Alternating Series Error Bound

For a series that alternates with sizes decreasing to 0, the sum lies within a_(n + 1) , the first omitted term, of S_n : |S − S_n| ≤ a_(n + 1) .

Practice 1: Implementing Mathematical ProcessesPractice 3: JustificationPractice 4: Communication and Notation

Part 1 · Hook

Why this matters

A partial sum is an estimate of a series' sum. For most series you cannot tell how good the estimate is, but for an alternating series you can, at a glance: the error is never bigger than the first term you left out.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which conditions does the alternating series test need for $\sum (-1)^n a_n$ with $a_n > 0$ ?

The $a_n$ decrease and approach 0
The $a_n$ approach 0
The $a_n$ decrease
$\sum a_n$ converges

Show the answer

Both: decreasing sizes that approach 0.

Correct: The $a_n$ decrease and approach 0:
The $a_n$ approach 0:
The $a_n$ decrease:
$\sum a_n$ converges:

2. What is $S_3$ for $\sum_{n=1}^{\infty} a_n$ ?

$a_1 + a_2 + a_3$
$a_3$
$a_1 + a_2$
$a_3 + a_4 + \cdots$

Show the answer

The third partial sum is the sum of the first three terms.

Correct: $a_1 + a_2 + a_3$ :
$a_3$ :
$a_1 + a_2$ :
$a_3 + a_4 + \cdots$ :

Part 4 · See it

See it first

Partial sums of 1 minus 1/4 plus 1/9 minus 1/16 and so on, for N = 1 to 8, zigzagging around a dashed line at pi squared over 12, about 0.822. A shaded band runs from S sub 4 = 115/144, about 0.7986, up to S sub 4 plus the next term, 1/25. The sum lies inside the band: the error of S sub 4 is about 0.0239, less than 1/25. — For $\sum \frac{(-1)^{n+1}}{n^2}$ , the sum lies between $S_4$ and $S_4 + a_5$ : the error of $S_4$ is less than $a_5 = \frac{1}{25}$ . LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

The partial sums of an alternating series zigzag around the sum with shrinking stepsthe sum always lies between two consecutive partial sums, $S_n$ and $S_{n+1}$
$S_{n+1}$ and $S_n$ differ by the next term, $a_{n+1}$ in sizethe distance from $S_n$ to the sum is at most $a_{n+1}$ : $|S - S_n| \le a_{n+1}$
The sign of the first omitted termtells the direction: if it is negative, $S_n$ overestimates the sum; if positive, it underestimates

Part 6 · Key ideas

Key ideas

If a series alternates and its sizes decrease to 0, then $|S - S_n| \le a_{n+1}$ : the error is at most the first term left out.
State the conditions every time; the bound means nothing without them.
Write the result as an inequality: "error $\le \frac{1}{720} < 0.002$ ". The error is at most the bound, not equal to it.
The sign of the first omitted term says whether $S_n$ is too big or too small.

Part 7 · Misconception

A common mistake

The wrong idea: The error of $S_3$ is at most $a_3$ , the last term added.

What actually happens: The bound is the first term left out, $a_4$ . The last term added is already part of $S_3$ .

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

The terms of an alternating series

The series $S$ $= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n \cdot 2^n}$ $= \frac{1}{2} - \frac{1}{8}$ $+ \frac{1}{24} - \cdots$ and the sizes of its terms
$n$	$1$	$2$	$3$	$4$	$5$	$6$
$a_n = \frac{1}{n \cdot 2^n}$	$\frac{1}{2}$	$\frac{1}{8}$	$\frac{1}{24}$	$\frac{1}{64}$	$\frac{1}{160}$	$\frac{1}{384}$

1. Find $S_3$ , the sum of the first three terms.

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$S_3$ $= \frac{1}{2} - \frac{1}{8}$ $+ \frac{1}{24}$ $= \frac{12 - 3 + 1}{24}$ $= \frac{10}{24}$ $= \frac{5}{12}$ .

Answer: $\frac{5}{12}$ ≈ 0.417

2. Use the alternating series error bound to find a number $E$ with $|S - S_3| \le E$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

The series alternates and its sizes $\frac{1}{n \cdot 2^n}$ decrease to 0, so the alternating series error bound applies: $|S - S_3| \le a_4 = \frac{1}{64}$ .

Answer: $\frac{1}{64}$ ≈ 0.016

3. Is $S_3$ an overestimate or an underestimate of $S$ ?

An underestimate: the terms keep getting smaller in size
An underestimate: $S_3$ ends with a positive term, $\frac{1}{24}$
Neither: $S_3$ equals $S$ to two decimal places
An overestimate: the next term, $-\frac{1}{64}$ , is negative

Show the answer

The partial sums zigzag around $S$ , and the next step, $-\frac{1}{64}$ , goes down. So $S < S_3$ : $S_3$ is an overestimate. (Indeed $S_3 \approx 0.4167$ and $S \approx 0.4055$ .)

An underestimate: the terms keep getting smaller in size: Shrinking terms do not decide the direction; the sign of the next term does.
An underestimate: $S_3$ ends with a positive term, $\frac{1}{24}$ : Ending with a positive term means $S_3$ overshot; the sum is below it.
Neither: $S_3$ equals $S$ to two decimal places: $S$ lies strictly between $S_3$ and $S_4$ , so $S_3$ overshoots it.
Correct: An overestimate: the next term, $-\frac{1}{64}$ , is negative: Right: the next step would go down, so $S$ is below $S_3$ .

4. Which statement correctly justifies that $S_4$ is within 0.01 of $S$ ?

$|S - S_4| = \frac{1}{160}$ , which is less than 0.01
The series alternates with sizes decreasing to 0, so $|S - S_4| \le \frac{1}{160} < 0.01$
The terms approach 0, so $S_4$ is close to $S$
$|S - S_4| \le \frac{1}{64} < 0.01$ , since $\frac{1}{64}$ is the fourth term

Show the answer

The alternating series error bound applies because the series alternates and $\frac{1}{n \cdot 2^n}$ decreases to 0. It gives $|S - S_4| \le a_5$ $= \frac{1}{160}$ $= 0.00625 < 0.01$ .

$|S - S_4| = \frac{1}{160}$ , which is less than 0.01: The bound gives "at most", not "equals"; the actual error is smaller than $\frac{1}{160}$ .
Correct: The series alternates with sizes decreasing to 0, so $|S - S_4| \le \frac{1}{160} < 0.01$ : Right: conditions, the bound, and the comparison with 0.01.
The terms approach 0, so $S_4$ is close to $S$ : Terms approaching 0 do not give a size; the bound $a_5$ with its conditions does.
$|S - S_4| \le \frac{1}{64} < 0.01$ , since $\frac{1}{64}$ is the fourth term: The fourth term is used in $S_4$ ; the bound is the first term left out, $\frac{1}{160}$ .

5. A student says the error in $S_{10}$ for $\sum_{n=1}^{\infty} \frac{1}{n^2}$ is at most $\frac{1}{121}$ , the first term left out. What is wrong?

The first term left out is $\frac{1}{100}$ , not $\frac{1}{121}$
The bound should be $\frac{1}{121} + \frac{1}{144}$ , the next two terms
Nothing: the argument is correct as written
The series does not alternate, so this bound does not apply

Show the answer

The alternating series error bound needs alternating signs. Here every omitted term is positive, so the error is $\frac{1}{121} + \frac{1}{144} + \cdots$ , which is more than $\frac{1}{121}$ (about $0.095$ ).

The first term left out is $\frac{1}{100}$ , not $\frac{1}{121}$ : $S_{10}$ uses $n = 1$ to $10$ , so the first omitted term is $\frac{1}{121}$ .
The bound should be $\frac{1}{121} + \frac{1}{144}$ , the next two terms: Adding two terms is still too small: the error is the sum of the whole tail.
Nothing: the argument is correct as written: The bound needs alternating signs; here every omitted term is positive.
Correct: The series does not alternate, so this bound does not apply: Right: with positive terms the error is the whole positive tail, larger than one term.

6. The first omitted term of an alternating series is $0.003$ , and the alternating series error bound applies. Which sentence is correct?

The error is exactly $0.003$
The error is at most $0.003$
The error is at least $0.003$
The error is about $0.0015$

Show the answer

The alternating series error bound says $|S - S_n| \le 0.003$ : the error is at most $0.003$ . Saying it "equals" $0.003$ loses the point.

The error is exactly $0.003$ : The actual error is less than or equal to the bound, usually less.
Correct: The error is at most $0.003$ : Right: the bound is a maximum.
The error is at least $0.003$ : The bound is an upper limit, not a lower one.
The error is about $0.0015$ : The bound does not estimate the error; it caps it.

Part 9 · Summary

Summary

For a series that alternates with sizes decreasing to 0, the sum lies within

a_{n+1}

, the first omitted term, of

S_n

|S - S_n| \le a_{n+1}

. State the conditions, compute the bound, and compare it with the required accuracy using "

\le \ldots <

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic