Lecture Notes on Series - Calculus of One Variable II | MTHSC 108, Exams of Mathematics

Download Lecture Notes on Series - Calculus of One Variable II | MTHSC 108 and more Exams Mathematics in PDF only on Docsity! Series Handout Tests for Convergence or Divergence Test Name IF THEN Geometric Series |r| < 1 (|r| ≥ 1) ∑∞ n=0 cr n converges (diverges) also, sn = c 1−r n 1−r and sn → c 1−r Tests for series with positive terms Integral Test ∫∞ N f(x) dx converges ∑∞ n=0 f(n) converges p-Series p > 1 (p ≤ 1) ∑∞ n=1 1/n p converges (diverges) Comparison Test an ≤ bn and ∑ bn converges ∑ an converges an ≥ bn and ∑ bn diverges ∑ an diverges Ratio Test an+1an → ρ and ρ < 1 (ρ > 1) ∑ an converges (diverges) Tests for series with terms of any sign Limit Test (nth term test) limn→∞ an 6= 0 ∑ an diverges Alternating Series Test ∑ (−1)nun with 1) un ≥ 0 ∑ (−1)nun converges 2) un ≥ un+1 and 3) un → 0 Absolute Convergence Test ∑ |an| converges ∑ an converges Behavior of Power Series Corollary to Theorem 18 The convergence of the series ∑ cn(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 with |x− a| > R but converges absolutely for x with |x − a| < R. The series may or may not converge at either of the endepoints x = a − R and x = a+R. 2. The series converges absolutely for every x (R =∞). 3. The series converges at x = a and diverges everywhere else (R = 0). Here, R is called the radius of convergence. The intervals of the form • a−R < x < a+R or (a−R, a+R), • a−R < x ≤ a+R or (a−R, a+R], • a−R ≤ x < a+R or [a−R, a+R), and • a−R ≤ x ≤ a+R or [a−R, a+R] are called intervals of convergence. 1