Note: Part (c) is worded in a way that might be confusing. It refers to the fact that, given a fixed whole number n, we only need the function f to be decreasing on [71, DC) to estimate 1?\". The question is asking whether the given 71 is such that f is decreasing on [71,, 00), and hence the Remainder Estimate Theorem applies. When applicable, the Remainder Estimate Theorem for the Integral Test allows us to estimate the sum 5' of a convergent infinite series 2 on using a partial sum and a corresponding improper integral. The goal of this exercise is for you to learn how to apply this theorem to estimate the sum of a convergentinfinite series, namely: E k k=1 8M3 71 Definition: For each integer n 2 1, let 371, = Z Gk = a1 + ' ' ' + on k:1 oo andlel Rn = Z ak=an+1+an+2+... k=n+l Thus. Sn denotes the nth partial sum of the series {the sum of its first 71 terms), and RR denotes the "remainder" of the series, which can be thought of as 00 the "infinite tail" of the series S = Z Gk. In other words, S = .971 + Rn. k:1 Theorem (Remainder Estimate Theorem for the Integral Test) Let n be an integer such that n 2 1. For each integer k 2 1, let ah, = f(k). If f(:l!) is a continuous, positive, decreasing function for all a: 2 n and the infinite series 2 an converges to S, then 00 f(z)dm g Rm. 5 [:0 f(m)d::. n+1 Do In particular, since Rn = S Sn: we have the following bounds on the exact sum 5' : Z \"k of the series: 15:] 00 oo Sn +f f(;c)da: S S 3 Sn +f f(:1:)da:. n+1 n m a) Consider f(:t':) = [3 . From below, select all correct statements: a: e C] The function f(:27) is continuous for all a: 2 1. C] For all a: 2 1, we havex) > 0 , that is, f(:1:) is postive for all x 2 1. [3 For all a: > 3, the function x) is decreasing. 00 C] The improper integral] f(:2:) do: is convergent. 1 is: b) For each integer is 2 1, define \"is = fUc) : Is the following statement true orfalse? etc/3 ' 0 F: By virtue of the Integral Test, we may conclude that the infinite series 2 is convergent. _ etc/3 kl O True 0 False 00 c) Does the Remainder Estimate Theorem for the Integral Test apply to Z for n = 5 ? k/3 k=l e 0 Yes O No CD By fully evaluating approriate improper integrals, find the lower L and upper U bounds on R5 for which the Remainder Estimate Theorem guarantees L E R5 S U. L : ab % \\/a_ |a| 11' 5mm) till 0 ab % a |a| 1r sin (a) m] o B FORMATTING: if the theorem applies, give exact expressions forL and U. Type your expressions using cafcul'ator notation, e.g., 811/5 is typed as e"(a/5). if the theorem does not appiy, enter 333