Question

(a) Starting with the geometric series X?, find the sum of the series η ΕΟ Σ ηχο – 1, 1x] <1. ΠΕ 1 (b) Find the sum of each of the following series. DO Σηχή, 1x <1 η = 1 η (i) Σ. (c) Find the sum of each of the following series. D) Σπίη – 1)x, Ix <1 ΠΕ 2 (i) Σ - η 57 ΠΕ 2 0 i) 22 = 1

Answer 1

formula - d (9-x)" n (a-xml vode 0-2 da -na-n) Infinite G.P s 1+kt q? + x3 = (a first number first Number Common ratio 1-X Tax 1-k

Q- G.M Series S E x" XI 1. n=o = It z + x² + x² I E lart 1-x - x, h=0 a). Diff both Side We respent to re 8 W d dx (ix) o swi hal non -) (-1) (1-x) 2 11 (1-x) 2

b) i) => q nox f (1-x)2 دن) nal n.x +1 0 + & (1-x) 2 h=1 MO 2 Σ και. +/ (1-x)2 2 DO tx Σ και.χ Am (1-x)? n=1 6) ii) h SMS § ls En.lt n=1 n=1 + 1/4 4 / (374)2 = 11 9 11-402 Now c) - i) Since 8W in-1 nx from equcil (1-x2 n=0 Diff respent to to x d dx E nx d hao dx (1-x)2

n-2 - 2 I n(n-1) a (-1) M 3 (1-x) 3 no 2 (1-2)3 n-2 2 otot E n(n-1) a 11 (1-x) 3 n=2 8 W 2 x n(n-1 2 2 (1-x) 3 n=2 E n(n-1). " 2 ri 2x4 3 (1-x) dhus Aus n=2 ii) - E 21 5h { n(n-1) わっ2 n=2 2 2x) (1-333 12 / 2 32

c). iii). iii) _ 2 2 n E + E E 25 2n E 24 n=1 ne) h=/ n2_n PW E + E nal 25 21 n=1 n n(n-1) ร 25 ༡༡ nal n =) = 0+ Ê n(n-1) ()" + Ŝ Enco" n=1 n=2 2 = 24 (1) 11- )? + 11-12 > y + 2 = 6 h