When applicable, the Remainder Estimate Theorem for the Integral Test allows us to estimate the sum S of a convergent infinite series _ an 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 convergent infinite series, namely: k K- 1 ek /2 Definition: For each integer n > 1, let Sn = >ak = aj + . .. +an K = 1 and let Rn I ak = ant1 + an+ 2 + ... k=n+1 Thus, Sn denotes the nth partial sum of the series (the sum of its first in terms), and Ron denotes the "remainder" of the series, which can be thought of a the "infinite tail" of the series S = > ax. In other words, S = Sn + Rn. K - 1 Theorem (Remainder Estimate Theorem for the Integral Test) Let in be an integer such that n > 1. For each integer k 2 1, let ax = f(k). If f(x) is a continuous, positive, decreasing function for all I > n and the infinite series _ an converges to S, then In + 1 f( z ) du E An f(z)dz. In particular, since Rn = S - Sn, we have the following bounds on the exact sum S =) ax of the series: k = 1 In+i f(2 ) dissent f(z) dz. a) Consider f(I) = - I . From below, select all correct statements: ex/2 The function f(z) is continuous for all z > 1. For all I 2 1, we havef(z) > 0, that is, f(x) is postive for all z 2 1. For all I > 2, the function f (z) is decreasing. Submit Assignment Quitk=1 Snt f(z) da 0, that is, f(x) is postive for all z > 1. For all z > 2, the function f (x) is decreasing. The improper integral / f(z) dx is convergent. b) For each integer k > 1, define ap = f(k) = - Is the following statement true or false? ek/2 By virtue of the Integral Test, we may conclude that the infinite series k - is convergent. k=1 ek/2 O True O False c) Does the Remainder Estimate Theorem for the Integral Test apply to> k - for n = 3? k=1 ek/2 O Yes O No d) By fully evaluating approriate improper integrals, find the lower _ and upper O bounds on Ro for which the Remainder Estimate Theorem guarantees L S R3