Math 214

Infinite series/series

The sum of the terms of an infinite sequence

Integral test

Given a positive, continuous, decreasing function f on [1, infty) and a_n = f(n), if the integral f(x)dx from 1 to infty converges/diverges then sum a_n converges/diverges

P-series

Sum of 1/n^P. Converges iff p>1

Comparison test

Given 0<=an<=bn… squeeze test

Given two positive series an, bn, lim an/bn =c, then 0<c<infty implies

both series converge or both series diverge

Alternating series

A series whose terms are alternatively positive and negative. an = (-1)^(n-1)bn where bn = |an|

Alternating series convergence test

An alternating series converges iff bn>0, bn is decreasing, and the limit of bn is 0.

Alternating series estimation

Given s as an alternating series and sn the nth partial sum, bn converges => |s-sn|<=b(n+1)