AP Calculus BC

Does the Sum of n/(n+1) Converge? No

Q: Does the sum of n/(n+1) converge?

No. The terms tend to 1, not 0, so the nth term test proves divergence.

The sum of n over n plus 1 diverges. Its terms approach 1, not 0, and a series whose terms fail to vanish cannot converge. This is the one situation where the nth term test settles the question outright.

\sum_{n=1}^{\infty}\frac{n}{n+1}

Diverges

Settled by the nth term test for divergence.

The nth term test, used correctly

Check the limit of the terms before anything else. It is the cheapest test and the only one that can end the problem in a single line.

\lim_{n \to \infty}\frac{n}{n+1} = 1 \neq 0

Terms settling near $1$ mean you are eventually adding roughly $1$ forever, so the partial sums grow without bound.

The test is one-directional

A nonzero limit PROVES divergence. A limit of zero proves nothing at all, and the harmonic series is the standing reminder of that.

The mistakes students make

Reading $\frac{n}{n+1}$ as tending to $0$ because both parts grow. Equal degrees give the ratio of leading coefficients, which is $1$ .
Reaching for the ratio or integral test first. Always check the terms first; here it finishes immediately.
Concluding convergence from the terms getting closer together. They approach $1$ , and adding numbers near $1$ forever diverges.

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 n/(n+1) converge?

No. The terms tend to $1$ , not $0$ , so the nth term test proves divergence.

Which test should I try first?

Always the nth term test. It is one limit, and when the limit is nonzero the problem is over.