Q13/095) Find coefficient of xⁿ in the expansion of ( 1 + 2x + 3x² + 4x³ + ........ inf. )^(1/2)....?

let y = ( 1 + 2x + 3x² + 4x³ + ........ inf. ...(1)
y converges for |x| < 1

now xy = x + 2x^2 + 3x^3 ....(2)

subtract (2) from (1)

y - xy = 1 + x + x^2 ... = 1/(1-x)

so y(1-x) = 1/(1-x)

or y = 1/(1-x)^2

so ( 1 + 2x + 3x² + 4x³ + ........ inf. )^(1/2)..= 1/(1-x) = 1 + x + x^2 ...

so coefficient of x^n = 1 for all n