Testing Σ(1/n²)

Calculus 2 · Axiom Academy

Apply the Integral Test step-by-step to show this p-series converges, then bound the remainder. Use the Integral Test to determine whether the series converges or diverges, and estimate the error in approximating its sum with the first 10 terms. Nice work! You applied the Integral Test end-to-end to settle convergence and bound the error. Here's what we learned: The Integral Test: if f(x) is continuous, positive, and decreasing on , then and either both converge or both diverge. Three key conditions: always verify continuous, positive, and decreasing before applying the test. Evaluating improper integrals: convert to a limit, . Remainder estimates: the integral bounds give , so we can bound the error in a partial sum. Convergence conclusion: since (finite), the series converges by the Integral Test. This technique works for many p-series and similar forms. The series actually converges to , a famous result of Euler — but remember, the Integral Test tells us that the series converges, not the exact sum (the integral value 1 is not the sum).

This is the written version of the interactive lesson above. See the full Calculus 2 course.