Test Chuỗi xen kẽ cho Sự Hội Tụ

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.

Partial sums of the alternating harmonic series zig-zag in toward ln 2; the error is at most the next term

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.

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 $b_n$.