AP Calculus BC

Does the Sum of 1/(2n+1) Converge? No

The sum of 1 over 2n plus 1 diverges. Limit comparison with 1 over n gives a ratio of one half, which is finite and positive, so the two series share a verdict. The harmonic series diverges, so this one does as well.

\sum_{n=1}^{\infty}\frac{1}{2n+1}

Diverges

Settled by the limit comparison test.

Halving the terms changes nothing

\lim_{n \to \infty}\frac{\frac{1}{2n+1}}{\frac{1}{n}} = \lim_{n \to \infty}\frac{n}{2n+1} = \frac{1}{2}

Any finite positive ratio is enough; it does not have to be $1$ . So this series diverges alongside the harmonic series.

Scaling never fixes divergence

Multiplying every term by a positive constant leaves the verdict alone. Half of infinity is still infinity, which is the intuition behind why the limit only has to be finite and positive.

The mistakes students make

Expecting the ratio to be $1$ . Any finite positive value works.
Splitting $\frac{1}{2n+1}$ into $\frac{1}{2n} + \frac{1}{1}$ . There is no such rule.

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/(2n+1) converge?

No. Limit comparison with the harmonic series gives a ratio of $\frac{1}{2}$ , so it diverges.

Does the ratio have to equal 1?

No, only finite and positive. Any such value forces the two series to agree.