AP Calculus BC

Does the Sum of sin^2(n)/n^2 Converge? Yes

The sum of sin squared n over n squared converges absolutely. Direct comparison settles it: sin squared n never exceeds 1, so every term is at most 1 over n squared, and that p-series with p equal to 2 converges.

∑n=1∞sin⁡2nn2\sum_{n=1}^{\infty}\frac{\sin^{2}n}{n^{2}}

Converges

Settled by the direct comparison test.

The numerator is trapped

Whatever sin⁡n\sin n does as nn runs through the integers, and it does not settle down, its square is confined to a fixed range.

0≤sin⁡2n≤10 \le \sin^{2} n \le 1

Dividing through by n2n^{2}, which is positive, preserves both inequalities.

0≤sin⁡2nn2≤1n20 \le \frac{\sin^{2} n}{n^{2}} \le \frac{1}{n^{2}}

∑1n2\sum \frac{1}{n^{2}} is the p-series with p=2>1p = 2 > 1, so it converges. A series of non-negative terms lying below a convergent series converges as well.

Non-negative terms, so absolute convergence is free

A square is never negative, so this series has no sign changes at all. That separates it from ∑sin⁡nn2\sum \frac{\sin n}{n^{2}}, where the comparison has to be applied to ∣sin⁡nn2∣\left|\frac{\sin n}{n^{2}}\right| before it means anything. Here the terms are already equal to their own absolute values.

Bounded beats erratic

Nothing here depends on how sine behaves at the integers, and no description of that behaviour is needed. The single fact carrying the argument is that the numerator stays inside a fixed interval.

The mistakes students make

Oscillation makes students reach for machinery this series does not need.

  • Reaching for the ratio test. The ratio of consecutive terms has no limit, because sin⁡2n\sin^{2} n keeps jumping around, so the test returns nothing usable.
  • Claiming sin⁡2n→0\sin^{2} n \to 0, or that it tends to anything. It has no limit, and the comparison never asks it to have one.
  • Assuming oscillation means alternating, and applying the alternating series test. Every term here is non-negative, so nothing alternates.

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 sin^2(n)/n^2 converge?

Yes, absolutely, by direct comparison with ∑1n2\sum \frac{1}{n^{2}}.

Does it matter that sin(n) has no pattern?

No. The only fact used is sin⁡2n≤1\sin^{2} n \le 1. The individual values never enter the argument.

What does this series add up to?

The comparison gives the verdict, not the total. Outside geometric and telescoping series, a convergence test never produces a value, and no closed form for this total is expected in AP Calculus.