AP Calculus BC

Does the Sum of 3n^2/(n^2+1) Converge? No

Equal degrees pin the terms of 3n squared over n squared plus 1 at 3, the ratio of the leading coefficients. Terms that settle at 3 never fade to 0, and any nonzero term limit is fatal, so the nth term test rules on the spot: this series diverges.

∑n=1∞3n2n2+1\sum_{n=1}^{\infty}\frac{3n^{2}}{n^{2}+1}

Diverges

Settled by the nth term test for divergence.

Equal degrees fix the limit

Top and bottom are both degree 22, so divide through by n2n^{2}.

3n2n2+1=31+1n2\frac{3n^{2}}{n^{2}+1} = \frac{3}{1 + \frac{1}{n^{2}}}

What is left is the ratio of the leading coefficients.

lim⁡n→∞3n2n2+1=31=3\lim_{n \to \infty}\frac{3n^{2}}{n^{2}+1} = \frac{3}{1} = 3

The terms creep up towards 33 rather than fading away, so the series diverges by the nth term test.

This test can never prove convergence

The nth term test runs one way only. If lim⁡an≠0\lim a_{n} \neq 0, the series diverges. If lim⁡an=0\lim a_{n} = 0, the test reports nothing whatsoever and a second test has to do the work.

lim⁡n→∞1n=0but∑n=1∞1n diverges\lim_{n \to \infty}\frac{1}{n} = 0 \quad \text{but} \quad \sum_{n=1}^{\infty}\frac{1}{n} \text{ diverges}

The harmonic series is the standard counterexample. Terms tending to 00 is necessary for convergence and nowhere near enough for it.

The mistakes students make

These show up in almost every set of scripts.

  • Claiming the limit is 00 because the denominator looks bigger. The degrees are equal, so the limit is 33.
  • Turning a limit of 00 into a claim of convergence. The nth term test cannot deliver that verdict for any series.
  • Comparing with ∑1n2\sum \frac{1}{n^{2}} on the strength of the n2n^{2} below the line. The numerator grows just as fast, and it is what settles the size of the term.

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 3n^2/(n^2+1) converge?

No. The terms tend to 33, so the nth term test gives divergence.

Can the nth term test prove that a series converges?

Never. It proves divergence only. A term limit of 00 leaves the question completely open.

What is the limit of 3n^2/(n^2+1)?

It is 33, the ratio of the leading coefficients, because numerator and denominator have the same degree.