National T

Notes

•Introduction This worksheet from National Taiwan University’s Calculus 4 (Class 01-09, Year 113) explores tests for series convergence without explicitly calculating partial sums. These tests also help estimate series sums. The first is the Integral Test, which involves improper integrals. •Integral Test Suppose \(f\) is continuous, positive, decreasing on \([1, \infty)\), and \(a_n = f(n)\). The series \(\sum_{n=1}^\infty a_n\) converges if and only if the improper integral \(\int_1^\infty f(x) dx\) converges. Specifically: (1) If \(\int_1^\infty f(x) dx\) converges, then \(\sum_{n=1}^\infty a_n\) converges. (2) If \(\int_1^\infty f(x) dx\) diverges, then \(\sum_{n=1}^\infty a_n\) diverges. This test helps analyze the convergence of p-series, which are of the form \(\sum_{n=1}^\infty \frac{1}{n^p}\), where \(p \in \mathbb{R}\). •Exercise 1 Consider \(\sum_{n=1}^\infty \frac{1}{n^2}\). (a) Find a continuous, positive, decreasing function \(f(x)\) such that \(f(n) = \frac{1}{n^2}\). (b) Check whether \(\int_1^\infty f(x) dx\) converges. (c) Use the Integral Test to determine if the series converges. •Exercise 2 We know the harmonic series \(\sum_{n=1}^\infty...