AP Calculus BC

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

Q: How do I spot the comparison quickly?

Subtract degrees: 2 - 1 = 1, so it acts like \frac{1}{n^{1}}, the divergent boundary case.

The sum of n over n squared plus 1 diverges. For large n the term behaves like n over n squared, which is 1 over n, so limit comparison with the harmonic series applies and the ratio is 1. The harmonic series diverges, so this one does too.

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

Diverges

Settled by the limit comparison test.

Compare with the dominant behaviour

Keep only the highest power on top and bottom to see what the term really looks like far out.

\frac{n}{n^{2}+1} \approx \frac{n}{n^{2}} = \frac{1}{n}

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

A finite positive ratio means the two series agree, and the harmonic series diverges.

Degree difference is the quick read

For a rational term, subtract the degrees: numerator degree 1 minus denominator degree 2 gives an effective p of 1, the boundary case, which diverges. Anything with a degree gap greater than 1 converges.

The mistakes students make

Concluding convergence because the terms tend to $0$ . They do, at exactly the harmonic rate, which is not fast enough.
Comparing with $\frac{1}{n^{2}}$ . The degree gap is $1$ , not $2$ .

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^2+1) converge?

No. It behaves like the harmonic series, which diverges.

How do I spot the comparison quickly?

Subtract degrees: $2 - 1 = 1$ , so it acts like $\frac{1}{n^{1}}$ , the divergent boundary case.