find the power series representation of the function fx 32x5

Solution

Start with the power series representation of arctan: arctan (x) = Int [1/ (1 + x^2)] dx =Int [1/ (1 - (-x)^2 ) ] dx = Int [1 - x^2 + x^4 - x^6 + .....] dx =[x - (x^3 /3) + (x^5 /5) - (x^7 /7) + .....] So, arctan (2x) = [(2x) - ( (2x)^3 /3) + ( (2x)^5 /5) - ( (2x)^7 /7) + .....] =[2x - ( 8x^3 /3) + ( 32x^5 /5) - ( 128x^7 /7) + .....] 5x arctan (2x)= [10x^2 - ( 40x^4 /3) + ( 160x^6 /5) - ( 640x^8 /7) + .....] c(0) = 10 c(1) = (-40/3) c(2) = (160/5) c(3) = (-640/7) c(4) = (2560/9)