calculus series tests cheat sheet, Cheat Sheet of Calculus

Download calculus series tests cheat sheet and more Cheat Sheet Calculus in PDF only on Docsity! 1. Convergence and Divergence Tests for Series Test When to Use Conclusions Divergence Test for any series ∞∑ n=0 an Diverges if lim n→∞ |an| 6= 0. Integral Test ∞∑ n=0 an with an ≥ 0 and an decreasing ∫ ∞ 1 f(x)dx and ∞∑ n=0 an both converge/diverge where f(n) = an. Comparison Test ∞∑ n=0 an and ∞∑ n=0 bn ∞∑ n=0 bn converges =⇒ ∞∑ n=0 an converges. if 0 ≤ an ≤ bn ∞∑ n=0 an diverges =⇒ ∞∑ n=0 bn diverges. Limiting Comparison Test ∞∑ n=0 an, (an > 0). Choose ∞∑ n=0 bn, (bn > 0) if lim n→∞ an bn = L with 0 < L < ∞ ∞∑ n=0 an and ∞∑ n=0 bn both converge/diverge if lim n→∞ an bn = 0 ∞∑ n=0 bn converges =⇒ ∞∑ n=0 an converges. if lim n→∞ an bn = ∞ ∞∑ n=0 bn diverges =⇒ ∞∑ n=0 an diverges. Convergent test ∞∑ n=0 (−1)nan (an > 0) converges if for alternating Series lim n→∞ an = 0 and an is decreasing Absolute Convergence for any series ∞∑ n=0 an If ∞∑ n=0 |an| converges, then ∞∑ n=0 an converges, (definition of absolutely convergent series.) Conditional Convergence for any series ∞∑ n=0 an if ∞∑ n=0 |an| diverges but ∞∑ n=0 an converges. ∞∑ n=0 an conditionally converges For any series ∞∑ n=0 an, there are 3 cases: Ratio Test: Calculate lim n→∞ ∣∣∣an+1 an ∣∣∣ = L if L < 1, then ∞∑ n=0 |an| converges ; Root Test: Calculate lim n→∞ n √ |an| = L if L > 1, then ∞∑ n=0 |an| diverges; if L = 1, no conclusion can be made. 2. Important Series to Remember Series How do they look Conclusions Geometric Series ∞∑ n=0 arn Converges to a 1− r if |r| < 1 and diverges if |r| ≥ 1 p-series ∞∑ n=1 1 np Converges if p > 1 and diverges if p ≤ 1