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Home Infinite Series Absolute and Conditional Convergence Comparison Tests for Absolute Convergence  
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Comparison Tests for Absolute Convergence
The comparison tests for positive term series give us tests for absolute convergence. COMPARISON TEST
LIMIT COMPARISON TEST
RATIO TEST
PROOF (i) Choose b with L < b < 1. By the ε, N condition, there is an N such that all the ratios are less than b. Therefore with c = a_{N}. a_{N+l} < cb, a_{N} _{+} _{2} < cb^{2},...,a_{N} _{+} _{n} < cb",.... The geometric series converges, so by the Comparison Test, the tail converges. Therefore the absolute value series converges. (ii) By the ε, N condition there is an N such that the ratios are all greater than one. Therefore a_{N} < a_{N+1} < ... < a_{N+n} < .... It follows that the terms a_{n} do not converge to zero, so the series diverges. The Ratio Test is useful even for positive term series, and is often effective for series involving n! and a^{n}.


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