Summary of Series Tests 2

Summary of Series Tests 2 - Summary of Convergence Tests...

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Unformatted text preview: Summary of Convergence Tests for Infinite Series Test Series Converges if Diverges if Comment lim an = 0 cannot be used to show convergence ∞ Divergence an n→∞ n=1 ∞ arn |r| < 1 |r| ≥ 1 1 np n=1 p>1 Sum: S = n=0 ∞ p-Series ∞ Integral ∞ ∞ f (x) dx an an = f (n) ≥ 0 ∞ ∞ and n=1 bn diverges 0 ≤ bn ≤ a n n=1 bn diverges an >0 n→∞ bn an >0 n→∞ bn and and an n=1 ∞ lim n=1 (−1) n−1 bn an , bn positive Test fails if lim n→∞ an =0 bn ∞ bn converges ∞ an , bn positive n=1 lim ∞ and decreasing ∞ and converges Limit Comparison f is continuous, positive, 1 converges 0 ≤ an ≤ bn an Comparison f (x) dx 1 n=1 Alternating Series a 1−r p≤1 Geometric Series bn diverges or lim n→∞ n=1 bn+1 ≤ bn n=1 lim bn = 0 an =∞ bn bn positive n→∞ lim bn = 0 n→∞ ∞ ∞ |an | an Absolute Convergence n=1 ∞ Ratio cannot be used to show divergence n=1 an n=1 converges an+1 lim <1 n→∞ an lim n→∞ an+1 >1 an Test fails if lim n→∞ ∞ Root an n=1 lim an n→∞ 1/n <1 lim an n→∞ 1/n >1 an+1 =1 an Test fails if lim an n→∞ Comments: The following general guidelines are useful: 1. Does the nth term approach zero as n approaches infinity? If not, the Divergence Test implies the series diverges. 2. Is the series one of the special types - geometric, telescoping, p-series, alternating series? 3. Can the integral test, ratio test, or root test be applied? 4. Can the series be compared favorably to one of the special types? 1 1/n =1 ...
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