Concept preview

What you will be able to do

1. 01State the alternating series test: $\sum(-1)^n b_n$ (with $b_n>0$) converges if $b_n$ is eventually decreasing and $b_n\to 0$, and check both hypotheses on a given series.
2. 02Explain why the alternating harmonic series $\sum\frac{(-1)^{n+1}}{n}$ converges even though the harmonic series diverges, and why dropping either hypothesis breaks the guarantee.
3. 03Apply the alternating series remainder bound $|S-S_N|\le b_{N+1}$ to estimate the error when a partial sum approximates the full sum.

An alternating series is one whose terms switch sign every step. We write it as $\sum (-1)^n b_n$ or $\sum (-1)^{n+1} b_n$, where the $b_n$ are the sizes of the terms — the positive numbers left over once the sign factor $(-1)^n$ is stripped away. So $b_n = |a_n|$ is always positive by design, and the alternating pattern lives entirely in the $(-1)^n$ out front. The archetype is the alternating harmonic series $1-\tfrac12+\tfrac13-\tfrac14+\cdots$, where $b_n=\tfrac1n$.