AP Calculus BC

Does the Sum of 1/n^2 Converge? Yes

The sum of 1 over n squared converges. It is the p-series with p = 2, and any p greater than 1 converges. Its exact value is pi squared over 6, about 1.6449, a result known as the Basel problem, though the AP exam only asks whether it converges.

∑n=1∞1n2\sum_{n=1}^{\infty}\frac{1}{n^{2}}

Converges

sum=π26\text{sum} = \frac{\pi^{2}}{6}

Settled by the p-series test.

Why p = 2 converges

With p=2>1p = 2 > 1 the p-series rule gives convergence immediately. The integral test shows why: the matching improper integral is finite.

∫1∞dxx2=[−1x]1∞=1\int_{1}^{\infty}\frac{dx}{x^{2}} = \left[-\frac{1}{x}\right]_{1}^{\infty} = 1

A finite integral bounds the tail of the series, so the partial sums are increasing and bounded above, which forces convergence.

The sum is pi squared over 6

Euler found the exact value in 1735, settling what was then called the Basel problem.

∑n=1∞1n2=π26≈1.644934\sum_{n=1}^{\infty}\frac{1}{n^{2}} = \frac{\pi^{2}}{6} \approx 1.644934

Knowing it converges is not knowing the sum

Every convergence test answers yes or no and almost never produces the value. Only geometric and telescoping series routinely give an exact sum on the AP exam; this one is famous precisely because the value is hard.

The mistakes students make

  • Assuming a convergence test hands you the sum. Tests decide convergence; the value usually needs a separate idea.
  • Reaching for the ratio test, which gives L=1L = 1 and decides nothing.
  • Confusing ∑1n2\sum \frac{1}{n^{2}} with ∑1n\sum \frac{1}{n}. One converges and one does not, and the exponent is the only difference.

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/n^2 converge?

Yes, by the pp-series test with p=2>1p = 2 > 1.

What is its exact sum?

π26\frac{\pi^{2}}{6}, roughly 1.64491.6449. The AP exam asks only for convergence, not the value.

Why does the ratio test fail here?

It returns L=1L = 1, the inconclusive case. Pattern-matching the pp-series is faster and decisive.