Alternating Series Test for Convergence
Introduced| English |
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| alternating series/ˈɔːltəneɪtɪŋ ˈsɪəriːz/ |
Series whose signs flip
- Some series alternate sign: $+,-,+,-,\dots$, like $1-\tfrac12+\tfrac13-\tfrac14+\cdots$.
- These are alternating series 交错级数, and they have their own gentle convergence test.
- The sign flips let terms partly cancel, so alternating series converge more easily.
- One test with two simple conditions handles them.
The Alternating Series Test
- An alternating series $\sum(-1)^n b_n$ (with $b_n>0$) converges if both:
- 1. the terms decrease: $b_{n+1}\le b_n$, and
- 2. the terms shrink to zero: $\displaystyle\lim_{n\to\infty}b_n=0$.
- If both hold, the alternating series converges. Simple as that.
Shrinking alternating terms
If the positive part decreases to $0$, the sign flips let the partial sums close in — the series converges.
The Alternating Series Test requires the positive terms $b_n$ to be which?
Decreasing and limit $0$.
Why the flips help
- Each new term is smaller than the last and opposite in sign, so it partly undoes the previous one.
- The partial sums bounce, but by shrinking amounts — closing in on a limit.
- That's why $1-\tfrac12+\tfrac13-\cdots$ converges, even though $1+\tfrac12+\tfrac13+\cdots$ (harmonic) diverges.
- The alternation is what tames the harmonic terms.
The alternating harmonic series $\sum\tfrac{(-1)^{n+1}}{n}$...
$b_n=\tfrac1n$ decreases to $0$ → converges.
The sign ____ let the partial sums close in on a limit.
Alternation causes partial cancellation.
$\sum\tfrac{(-1)^n}{n}$ converges even though $\sum\tfrac1n$ diverges.
The alternation tames the harmonic terms.
Applying it cleanly
- Strip off the $(-1)^n$ and check the positive part $b_n$ for the two conditions.
- Is $b_n$ decreasing? Does $b_n\to0$? If yes to both, converges.
- If $b_n$ does not go to $0$, the series diverges (by the nth Term Test instead).
- The test is quick once you isolate $b_n$.
You check the two conditions on the positive part $b_n$ after removing $(-1)^n$.
Isolate $b_n$ first.
If $b_n\not\to0$ for an alternating series, it...
Terms not $\to0$ → diverges regardless of alternation.
Apply the two conditions to the positive terms $b_n$ (after removing the $(-1)^n$). Both are needed: decreasing and limit $0$. If $b_n\not\to0$, the series diverges (nth Term Test). And this test proves convergence of the alternating series; it says nothing about $\sum b_n$ or $\sum|a_n|$ (see absolute vs. conditional, lesson 10.9).
Does the alternating harmonic series $1-\tfrac12+\tfrac13-\tfrac14+\cdots$ converge?
- Positive part $b_n=\tfrac1n$. Decreasing? Yes, $\tfrac1{n+1}<\tfrac1n$. Limit $0$? Yes, $\tfrac1n\to0$.
- Both conditions hold, so the series converges by the Alternating Series Test.
- (Even though the plain harmonic series diverges.)
The Alternating Series Test: $\sum(-1)^n b_n$ (with $b_n>0$) converges if the terms $b_n$ are decreasing and $b_n\to0$. Check both on the positive part $b_n$. The alternation lets series like $\sum\frac{(-1)^n}{n}$ converge where the non-alternating version diverges.