University:
Santa Ana College
Course:
MATH 150 | Calculus for Biological, Management and Social Sciences
Academic year:

2023
Views:

425

Pages:

2
Author:

hekeldigzjbz

Review of Convergence Tests STATEMENT NAME Divergence Test If lim 0, then Za, diverges. Integral Test Let Zu be a series with positive terms, and let Ax) be the function that results when & is replaced by x in the formula for u. If fis decreasing and continuous for x > 1. then and Mel da both converge or both diverge. Comparison Test Let Sax and 26, be series with nonnegative terms such that baba b If be converges, then Sa, converges, and if Ed diverges, then Sh, diverges. Let Zu be a series with positive terms and suppose Ratio Test limp (a) Series converges if p < 1. (b) Series diverges if p 1 or p (c) No conclusion if p1. Let Ew, be a series with positive terms such that p= lim V Root Test (a) Series converges if p < 1. (b) Series diverges if p 1 or p (c) No conclusion if p = 1. Limit Comparison Test Let Za, and Eb, be series with positive terms such that If0p, then both series converge or both diverge. Alternating Series Test The series converge if (b) lim (0 COMMENTS If lim = 0. Sa may or may not converge. Use this test when fir) is easy to integrate. This test only applies to series that have positive terms. Use this test as a last resort; other tests are often easier to apply. This test only applies to series with nonnegative terms. Try this test when w, involves Factorials or 4th powers. Try this test when as involves Ath powers This is easier to apply than the comparison test, but still requires some skill in choosing the series Σε, for comparison. This test applies only to alternating series. It is assumed that 0 for all k Summary of convergence and divergence tests for series TEST SERIES Σα CONVERGENCE OR DIVERGENCE COMMENTS nth-term Diverges if lim, 4,0 Inconclusive if lim, a=0 Geometric series (1) Converges with sum Sifr<1 Useful for comparison tests if the ath term a, of a series is similar to ar p-series (i) Converges if p> 1 (ii) Diverges if 21 Useful for comparison tests if the nth term a, of a series is similar to 1/m² Integral (ii) Diverges if p≤1 (0) Converges if f f(x) dx converges (ii) Diverges if f f(x) ds diverges The function / obtained from (n) must be continuous. positive, decreasing, and readily integrable. The comparison series b, is often a geometric series or a p-series. To find b, in (iii), consider only the terms of a, that have the greatest effect on the magnitude. Comparison Σα Σα a>0,b, 0 (i) IIb, converges and a, b, for every n, then a, converges. (ii) Ib, diverges and a, b, for every n. then ∑a, diverges. (iii) If lim, (a/b)>0, then both series converge or both diverge.. Ratio Σα If lim L(or), the series (i) converges (absolutely) if <1 Inconclusive ifㄥ=1 Useful if a, involves factorials or ath powers If a > 0 for every n, the absolute value sign may be disregarded. (ii) diverges if L> 1 (or oc) Root Σ If limL (or), the series (i) converges (absolutely) if L. <1 (ii) diverges if L> 1 (or) Inconclusive if L-I Useful if a, involves nth powers If a, 0 for every n. the absolute value sign may be disregarded. Alternating series Στα 4>0 Converges if a for every k and lima, 0 Applicable only to an alternating series Σ Σα Ia, converges, then a, converges. Useful for series that contain both positive and negative terms a.-f(n)

Math Cheat Sheet Convergence Tests

Related Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock