Alternating series and the error bound

Warm-up

Three quick questions from earlier lessons.

Discover

$1 - \tfrac{1}{2} + \tfrac{1}{3} - \tfrac{1}{4} + \dotsb$ : the partial sums go $1, 0.5, 0.83, 0.58, 0.78, \dots$ . Do they settle?

Yes
No, because $\sum \tfrac{1}{n}$ diverges

Key idea

Alternating signs help convergence, and give an error bound.

Alternating series test: if

b_n > 0

decreases to 0, then

\sum (-1)^{n+1} b_n

converges, and

|S - S_N| \le b_{N+1}

(the first term left out). If

\sum |a_n|

converges, the series converges absolutely; if only

\sum a_n

does, conditionally.

Worked example

$\displaystyle\sum \dfrac{(-1)^{n+1}}{n^3}$ with 3 terms.

S_3 = \tfrac{197}{216} \approx 0.912

1 − 1/8 + 1/27.

|S - S_3| \le \tfrac{1}{64}

First omitted term.

Absolutely convergent

∑1/n³ converges.

Your turn, guided

Your turn: the error bound for $\displaystyle\sum \dfrac{(-1)^{n+1}}{n}$ after 4 terms.

\tfrac{1}{5}

b₅.

Practice ★

Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 7 }{n}$ converge?
1. Yes, conditionally: the terms alternate, shrink and tend to 0
2. No: ∑ 7/n diverges
3. Yes, absolutely

Find the partial sum $S_3$ of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 1 }{n}$ .
Find the partial sum $S_3$ of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 7 }{n}$ .

Watch out!

Practice ★★

$S = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} }{n^2}$ is approximated by its first 8 terms. Give the alternating-series bound on the error.
$S = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} }{n^2}$ is approximated by its first 2 terms. Give the alternating-series bound on the error.

Boss problem. $S = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} 1 }{n^2}$ .

Exit ticket

Three questions to check you have got it.

$S = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} }{n^2}$ is approximated by its first 8 terms. Give the alternating-series bound on the error.
$S = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} }{n^2}$ is approximated by its first 2 terms. Give the alternating-series bound on the error.
$S = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} }{n^2}$ is approximated by its first 6 terms. Give the alternating-series bound on the error.

Summary card

Alternating series

$b_n$ decreasing to 0: converges.
Error after $N$ terms $\le b_{N+1}$ .
Absolute vs conditional: test $\sum |a_n|$ .

Watch out: the bound uses the first omitted term.

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