1. The Integral Test: Let {anL 3,021 be a positive sequence and suppose that f (T) is an eventually continuous, positive, and decreasing function with f0?) = a," for all n S N for some integer N. Then, 00 Z a}, and /00 f(:r)d:r . N k:=N either both converge or both diverge. (a) Underline the hypothesis of the above theorem. Box the conclu- sion. Use this to create a checklist of things that must be true to apply the integral test. (b) What does it mean for that integral to converge? What kind of integral is it? Hint: Think back to 21B. (c) Consider the below image. Explain in your own words how this image relates to the integral test. Think about how and When the integral represents area under the curve. (L, r, I. ~ --'_ H. l 2 3 4 F J (d) What goes wrong if the sequence is increasing? What about if the sequence has negative terms? (e) True or false: if the integral test applies, then (1.: 00f(:1:)drr