AP Calculus BC / Unit 10: Infinite Sequences and Series
10.10 Alternating Series Error Bound
Notes
For an alternating series meeting the test, the error after $n$ terms is at most the first unused term:
$$|S-S_n|\le b_{n+1}$$
Practice
Using 3 terms of $\displaystyle \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}$, the error is at most:
The next term is $\displaystyle \frac{1}{4}$.
How many terms of $\displaystyle \sum\frac{(-1)^{n+1}}{n^2}$ guarantee error under 0.01?
Need $\displaystyle \frac{1}{(n+1)^2}<0.01$, so $n+1>10$, giving $n=10$.
The alternating series error bound works only when:
Terms must decrease to 0.
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