SeriesTests3 - Series Tests Complete Summary Standard...

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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 the “Standard Series” or one that can be handled with the integral test.
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