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DIRECT COMPARISON TEST is presented in $5.4 of the OpenStax text. DIRECT COMPARISON TEST: Let \\ a, and \\ b,, be infinite series with positive terms. n=1 Suppose we know the behavior of > ba- n=1 (1) If >b,, converges and an S b, for n 2 N, then > an converges. n=1 (2) If > b,, diverges and an 2 b,, for n 2 N, then > a,, diverges. Our objective is to determine the behavior of given series. We generally represent the given series by > an. When we apply the Direct Comparison Test, we must come up with a new series to which to compare the given series, and we must know how this new series behaves. We refer to this new series as the contrived series, and we represent it by > b,. Once we n=1 have the given series and the contrived series, we can apply the Direct Comparison Test. In problems 1 - 4, both the given series and the contrived series (the series to which to compare the given series) are presented. Apply the Direct Comparison Test to determine whether the given series converges or diverges. 1. Given series: n' + n+ 1' ; contrived series: n= 1 712 1= 1 Converges 5 2. Given series: IM : M 2 + an; contrived series: Converges 3. Given series: M ; contrived series: 1=2 n- Vn Diverges 4. Given series: cos n 712 + 1 : ; contrived series: IM : 1=1 Converges