Series Handout Tests for Convergence or Divergence Test Name IF THEN Geometric Series jrj< 1 (jrj 1) P1n=0crn converges (diverges) also, sn = c1 rn1 r and sn! c1 r Tests for series with positive terms Integral Test R1N f(x) dx converges P1n=0f(n) converges p-Series p> 1 (p 1) P1n=1 1=np converges (diverges) Comparison Test an bn and Pbn converges Pan converges an bn and Pbn diverges Pan diverges Ratio Test an+1an ! and < 1 ( > 1) Pan converges (diverges) Tests for series with terms of any sign Limit Test (nth term test) limn!1an6= 0 Pan diverges Alternating Series Test P( 1)nun with 1) un 0 P( 1)nun converges 2) un un+1 and 3) un!0 Absolute Convergence Test Pjanjconverges Pan converges Behavior of Power Series Corollary to Theorem 18 The convergence of the series Pcn(x a)n is described by one of the following three possibilities: 1. There is a positive number R such that the series diverges for x withjx aj>R but converges absolutely for x with jx aj< R. The series may or may not converge ...