AP Calculus BC

Does the Sum of (-1)^n/(n+1) Converge?

The sum of negative 1 to the n over n plus 1 converges CONDITIONALLY. The magnitudes decrease to 0 so the alternating series test applies, but the absolute values give a shifted harmonic series, which diverges.

∑n=1∞(−1)nn+1\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n+1}

Converges

Settled by the alternating series test, and only conditionally.

A shifted alternating harmonic series

With bn=1n+1b_{n} = \frac{1}{n+1}, both conditions of the alternating series test hold: the magnitudes decrease and tend to 00.

Shifting the index cannot change convergence, since dropping or adding finitely many terms alters the sum but never the verdict.

Only the tail decides

Two series that agree from some point onward always share a verdict. That is why a shift, a reindex, or removing the first thousand terms is always safe when you are only asked whether it converges.

Conditional, because the absolute series diverges

∑n=1∞1n+1=12+13+14+⋯=∞\sum_{n=1}^{\infty}\frac{1}{n+1} = \frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots = \infty

That is the harmonic series missing only its first term, so it still diverges.

The mistakes students make

  • Thinking the shift makes the absolute series converge. Removing one term never changes a verdict.
  • Claiming absolute convergence because the terms alternate.

Not sure which test a series wants?

The Convergence Test Chooser walks the decision in order: nth term first, then geometric and p-series pattern matching, then alternating structure, then the ratio test, and finally the comparison family.

Frequently asked questions

Does the sum of (-1)^n/(n+1) converge?

Yes, conditionally, by the alternating series test.

Does shifting the index matter?

Not for convergence. Finitely many terms change the sum but never the verdict.