comparison_test - (12.4 The Comparison Tests Often a series...

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(12.4) The Comparison Tests Often, a series will resemble a simpler series whose convergence is known. This may enable us to determine if a series converges. An example is the series n =1 1 1 + 2 n . We notice that 1 1 + 2 n 1 2 n , so we know that n =1 1 1 + 2 n n =1 1 2 n . Since n =1 1 2 n = 1, we conclude that n =1 1 1 + 2 n 1. So we are assured that this series converges. (To what, we don’t know. It converges to some number 1. The partial sum with thirty terms adds up to approximately 0 . 7645.) The comparison test. Suppose 0 b n a n for all n , and we know that n =1 a n is convergent. Then so is n =1 b n . Suppose 0 a n b n and we know that n =1 a n is divergent. Then so is n =1 b n . Example. n =1 n n 3 + 1 . Notice n n 3 + 1 n n 3 = 1 n 2 . Since n =1 1 n 2 is convergent, the series n =1 n n 3 + 1 is also convergent. Example. n =2 2 n n 2 - 1 . Notice 2 n n 2 - 1 2 n n 2 = 2 n 1 n . So n =2 2 n n 2 - 1 n =2 1 n = . So the series n =2 2 n n 2 - 1 diverges.
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