SeriesTests3 - Series Tests — Complete Summary Standard...

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Unformatted text preview: Series Tests — Complete Summary Standard Series 1. Geometric Series ∞ X n =0 Ar n = A + Ar + Ar 2 + ··· = ( A 1- r if | r | < 1 diverges if | r | ≥ 1 2. p-Series ∞ X 1 n p converges if and only if p > 1 (e.g. ∑ 1 n 2 converges, ∑ 1 √ n diverges). 3. Constant Series ∑ ∞ c = c + c + c + ··· diverges (unless c = 0) 4. Exponential Series ∞ X n =1 x n n ! = e x (converges for any x by the ratio test). Our Tests 1. Integral Test: If f ( x ) is a continuous, non-negative, decreasing function, then ∞ X n =1 f ( n ) converges ⇐⇒ Z ∞ 1 f ( x ) dx is finite . 2. Comparison Test: If 0 ≤ a n ≤ b n for all large n , then ( ∑ b n converges = ⇒ ∑ a n converges ∑ a n diverges = ⇒ ∑ b n diverges 3. Limit Comparison Test: If a n ,b n ≥ 0 and lim n →∞ a n b n = L with L 6 = 0 or ∞ then ∑ a n and ∑ b n either both converge or both diverge. This makes precise the intuition that “ a n ≈ Lb n for large n ”. To apply it, take ∑ b n to be one of...
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This note was uploaded on 05/12/2008 for the course MATH 133 taught by Professor Wei during the Spring '07 term at Michigan State University.

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SeriesTests3 - Series Tests — Complete Summary Standard...

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