Alternating series: checking magnitudes and limits

Understand the idea

An alternating series can converge through cancellation even when the series of term magnitudes diverges. The alternating-series test supplies a sufficient condition: the positive magnitudes decrease eventually and approach zero.

Separate the sign pattern from the magnitude $\displaystyle b_n$ . Check that $\displaystyle b_n$ is positive, eventually nonincreasing, and tends to zero. A derivative of a related real-variable function or a direct inequality can establish decreasing behavior.

If the magnitudes do not approach zero, the original terms cannot approach zero and the series diverges. If monotonicity is not established, this particular test may be inconclusive; failure to verify a sufficient condition does not by itself prove divergence.

\sum(-1)^{n-1}b_n\text{ converges if }b_n>0,\ b_n\downarrow0\text{ eventually}

A method you can apply

Write the terms as an alternating sign times a positive magnitude.
Verify eventual decrease and the zero limit of the magnitudes.
Conclude convergence only after checking both conditions; investigate absolute convergence separately when needed.

Worked example

Determine whether $\displaystyle \sum_{n=1}^{\infty}(-1)^{n-1}/(n+2)$ converges.

The magnitude is $\displaystyle b_n=1/(n+2)>0$ .
Since the denominator increases, $\displaystyle b_{n+1}<b_n$ .
The magnitude tends to zero, so the alternating-series conditions hold.

The series converges. Its absolute-value series is a tail of the harmonic series, so this convergence is conditional.

Try it yourself

Does $\displaystyle \sum_{n=1}^{\infty}(-1)^n n/(n+1)$ converge?

Show solution and reasoning

No. The magnitudes approach $\displaystyle 1$ , so the signed terms do not approach zero. The nth-term divergence test rules out convergence despite the alternating signs.