Question

34. [-12 Points] DETAILS SCALCET8 11.9.501.XP. Find a power series representation for the function. (Give your power series representation centered at x = 0.) 9 f(x) = 16 + x 00 f(x) n = 0 Determine the interval of convergence. (Enter your answer using interval notation.)

Answer 1

Solution f(x) = 9 at no 16+2 Divide Numeratos & Denominator by 16, to get 8(9)= (%) i +(2) Note that the sum of geometer series with initial Term a & Common satio o See s={agna a 1=0 a 1-8 The given function can be interpreted as a 1--) 옳 Therefore we can say that flas is a sum of a geometric series with initial term azq & Common satio Therefore, fo- 8= -2 16 f(x)= {ax = . .)(49)". 2. (96030 3941"x" no 16 2.).2" no n+1 $100 This r the power series representation of flor). We know that the geometri series converges where 101-12 41 171 ) 111 <16 So, Interival of convergence in 1-16,16) Radies of convergence À 16