AP Calculus BC

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

Q: Does the plus one inside the square root change anything?

Not the verdict. Limit comparison with 1/\sqrt{n} gives a limit of 1, so the two series converge or diverge together. It changes the sum, not the classification.

The series converges conditionally. The sizes one over the square root of n plus one decrease to zero, so the alternating series test applies. Without the signs it becomes a p-series with p = one half, which diverges.

\sum_{n=1}^{\infty}\frac{(-1)^{n}}{\sqrt{n+1}}

Converges

Settled by the alternating series test, and only conditionally.

The two conditions

The sizes $b_n = \frac{1}{\sqrt{n+1}}$ decrease because the denominator strictly increases, and they tend to 0 because the square root grows without bound. Both conditions hold from the first term, so the alternating series test proves convergence.

Decay is slow: at $n = 10000$ the size is still about $0.01$ . Slow decay is fine for the test, which asks only that the sizes decrease to zero, not how quickly.

The absolute series, and the shift that does not matter

Removing the signs gives $\sum \frac{1}{\sqrt{n+1}}$ . The shift by one changes which terms appear but not the tail behaviour: this is the p-series with $p = 1/2$ starting one place later, and $p = 1/2 \le 1$ means divergence.

More formally, limit comparison against $\sum 1/\sqrt{n}$ gives $\lim \sqrt{\frac{n}{n+1}} = 1$ , a finite nonzero limit, so the two series share a verdict. Shifts and additive constants inside a root never change convergence.

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 plus one inside the square root change anything?

Not the verdict. Limit comparison with $1/\sqrt{n}$ gives a limit of 1, so the two series converge or diverge together. It changes the sum, not the classification.

How accurate is a partial sum of this series?

The alternating series error bound applies: truncating after n terms leaves an error no bigger than the next size, $1/\sqrt{n+2}$ . Reaching three decimal places needs about a million terms.