10.10 Alternating Series Error Bound

If an alternating series satisfies the conditions of the alternating series test, then the error bound for using the nth partial sum ${S_n}$ to approximate the full sum S is ${|S-S_n| \le b_{n+1}}$ where ${b_n}$ is the positive term size. This means the magnitude of the error is at most the first omitted term. This is extremely useful because it lets you estimate how accurate a partial sum is without knowing the exact value of the infinite series.

You should use this result only when the series is alternating with decreasing positive terms that go to 0. Then the partial sums bounce above and below the true sum, and the error gets trapped by the next term. If you are asked how many terms are needed for a desired accuracy, you solve the inequality ${b_{n+1}<\text{tolerance}}$. This is one of the most practical convergence results in the unit because it turns a convergence theorem into a numerical estimate tool.