Question

6. Show that each of the following series converge on the indicated interval and that the derivative of the sum can be obtained by term-by-term differentiation of the series. 岚(1 + kx)2. k! k=0

Answer 1

Due to indicated that question a is required a) Given that Sigma_k = 1^infinity 1/(1 + k x)^2, x elementof (0, infinity) Sigma_k = 1^infinity 1/(1 - k x)^2 By ratio test lim_x rightarrow infinity x(1 + k^2 z^2 + 21(x) /(1 + (k +) x)^2 lim_x rightarrow infinity K^2(1/k^2 + x^2 + 2 x/x)/k^2()1/k + (1/k + 1/2)^2 = x^2/x = x = 0 and x > 0 is the indicated interval of w_n vergence \r\n Now f(x) = sigma_i = 1^infinity 1/(1 + k x)^2 f(x) = 1/(1 + x^2) + 1/(1 + 2 x)^2 + 1/(1 + 3 x)^2 ..... 1/(1 + k x)^2 f(x) = (1 + x+^- 2 + (1 + 2 x)^- 2 + (1 + 3 x)^- 2 .... (1 + k x)^- 2 now differentiable above w r to x f'(x) = -8(1 + x)^- 3 - 2(1 + 2 x)^- 3 - 2 - 2(x + 3 x)^- 3 .... 2(1 + k x)^-03 k f'(x) = -2/(1 + x)^3 - 4/91 + 2 x)^3 - 6/(1 + 3 x)(^3 ..... 2 k/(1 + k x)^3