SeriesConvergenceTests - Calculus 107-452 Tests for Convergence n1 n=1 ar Proposition(Geometric Series Test If a > 0 and 1 > |r| then arn1 = n=1 If |r|

# SeriesConvergenceTests - Calculus 107-452 Tests for...

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Calculus 107-452 Tests for Convergence Proposition ( Geometric Series Test ) . If a > 0 and 1 > | r | , then n =1 ar n - 1 converges and X n =1 ar n - 1 = a 1 - r If | r | ≥ 1 , then n =1 ar n - 1 diverges. Proposition ( n th term test ) . If lim a n 6 = 0 , then X n =1 a n diverges Proposition ( The Integral Test ) . If a n is positive, and there exists a function f such that f ( n ) = a n and (i) f is decreasing. (ii) f is continuous. (iii) f is positive then X n = k a n and Z k f ( x ) dx either both converge or both diverge. Proposition ( Direct Comparison Test ) . If a n and b n are two sequences of nonnegative numbers such that a n b n , then X n =1 a n X n =1 b n So: If n =1 a n diverges, then n =1 b n diverges, and If n =1 b n converges, then n =1 a n converges. Proposition ( Limit Comparison Test ) . If a n and b n are two sequences of positive numbers, then if lim n →∞ a n b n n =1 b n diverges n =1 b n converges c n =1 a n diverges. n =1 a n converges. 0 ? n =1 a n converges.