Calculus question related to series please help asap!

Question

f(x) = summation from n equals 1 to infinity of the product of the quotient of the quantity n plus 1 and 4 to the n plus 1 power and x to the nth power with an interval of convergence, –4 < x < 4. Find exactly the value of the integral from 0 to 2 of f(x), dx. Your answer will be a positive integer.

Sorry about this format, ∨∨and here is a link to the question∨∨

https://media.cheggcdn.com/media%2F901%2F901195a5-738e-4df3-a09a-fe1991db7331%2FphpdKmQx6.png

Answer 1

f[x]=(1/4){ 1+2x/4 + 3x²/64+4x³/256+⋅⋅⋅}

Then

∫f(x)dx=(1/4)(x +x^2/4 +x^3/64 +x^4/256+⋅⋅⋅⋅) =(1/4)⋅[x+{(x/4)^2[1+(x/4)+(x/4)^2+(x/4)^3+⋅⋅⋅]}]

But 1+(x/4)+(x/4)^2+(x/4)^3+⋅⋅⋅=1/[1-(x/4)]

I think you should be able to take it from here