Let o an be a series with positive terms and let rn an11yan . Suppose that limnl` rn L

Chapter 11, Problem 46

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Let o an be a series with positive terms and let rn an11yan . Suppose that limnl` rn L , 1, so o an converges by the Ratio Test. As usual, we let Rn be the remainder after n terms, that is, Rn an11 1 an12 1 an13 1 (a) If hrn j is a decreasing sequence and rn11 , 1, show, by summing a geometric series, that Rn < an11 1 2 rn11 (b) If hrn j is an increasing sequence, show that Rn < an11 1 2 L