Question

1. When applying the Integral Test, it is not entirely correct to say that the value of the integral tells us nothing about the value of the infinite sum. We can use an improper integral to find upper and lower bounds for the value of the sum. That is, if f is a continuous, positive, decreasing function on [1, oo), and f(n) an, then a) Use pictures (and words) to show why these inequalities are true. (Hints: Do one inequality at a time. See p. 535.) b) Evaluating the p-series with p 2 (as shown in the historical note on p. 530) is well beyond the techniques of this course. However, formula (1) can he used to find an interval in which the sum is certain to fall. Do this for y 1 , and don't be too disappointed in the results. rn 」 c) So, the bounds found in part b are not very good. We can improve on the above inequalities as follows. o0 Explain why this is true as well, and use it to find a smaller interval in which 2 2 eliminating when we use formula (2) rather than (1). lies. Show in your pictures exactly what error we are Tl

Answer 1

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