Question

.... This is called the geometric series. 1. (a) Prove that 1+r+p2 + ... + -1 = (b) Use Riemann sums to calculate Pedro (Hint: At some point your Riemann sum may contain 1+e2/n + en + ... + 2(n-1)/n. What do you get if you set r = e2/n? You will probably have to use L'Hôpital's Rule at some point.)

Answer 1

Page No.: - 1 = (-1) (ruotonant I r"-1 - rutru-2 t...trti - r 2-1 tan-2 Given that love partition follows -oc < fra) = ea & the interval D E 10, 2] [0, 2] ar ... zen-1) 2 Consider intenals Ing Inj = {214+) , 2 is jen length Now of each Cosider internal upper is sum % a Mefis) 1 Jn. ;) to tff, 6) = - Un Z g=1 I function at end point en is a increasing - supremum occur the intenal. of zchn - Un = 3 [ gente torte 2 tentent.. inte + 5

22 I Let er IN - Un= ? I ltrtrto--trte ²17 = 2 Cr -1 te? = 2 | CC=1) + (e?-D a r ce ²D + - ce²) 7 rce? = 2(6²1) [ mro] n er-u an Un - = 2(621) x 2 ce ce re -12 Now, we finid nos limit at Un als

Date Page No.: that For lim n - e? 2'Hospital's sule - lim ne nx un nis 63 i tim - (2 +1) ein no as n = 2le² D x 1 = e ²1 Deimann sums K lisina e are = e ²