Consider the following family of seriesr one for each constant p > 0: $02) =50: 1 \":2 n[logn)1" For each N 2 2 we define the partial sum up to index Nanci the corresponding remainder as follows: N Styli?) : 22 m, RN09) : 3 3N0?)- (a) Express the following definite integral in terms of 0., b, andp (assume a > 1): fb d2: , a $(logm)1\" _ Usep : 3.2 in all parts below. Please enter your answers in calculator ready form. (b) Express the following definite integral in terms of a (assumea > 1): [0 aim _ a :c(log m)3-2 _ (c) The integral test provides a two-sided inequality of the following form for RN(3.2) when N : 20: 0 d 0 d f ng20(3.2)3f m 1 milog 5193'2 a2 milogwl3'2' Find the integers 0,1 and (12 above. [Use the smallest compatible value of a1 and the largest compatible value of (12-) a. I (I I 1 I 2 - (d) Computation reveals 520(32) : 2.093747. Use this fact, together with your earlier work, to complete this two- sided inequality involving the exact value of the series S(3.2): S 5(32) 5 I. (e) An experimentalist might express the twosided inequality in part (d) by writing S(3.2) : y, :l: s, where ,u is the midpoint of the interval and 5 is the distance from ,u to either endpoint. Translate your findings about the given series into experimentalists' notation: S(3.2) : :l