if -1<x <1 find the sum x + 2x^2 +3x^3 + ...... to infinity?

x + 2x^2+ 3x^3 + .. + kx^k + ...

= x(1 + 2x + 3x^2 + ... + kx^(k - 1) + ...)

= x d/dx (1 + x + x^2 + x^3 + ... + x^k + ...)

= x d/dx (1 / (1 - x))

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

= x / (1 - x)^2.

(Rewriting a power series using a term-by-term derivative preserves the radius of convergence, and 1 + x + x^2 + x^3 + .... + x^k + ... converges to 1 / (1 - x) when -1 < x < 1, so each step is justified.)

Try a taylor expansion of the form

(a+bx)*arctanh(x), and solve for a and b. If this doesn't work at first go, then extend the leading polynomial further, like

(a+bx+c*x^2 +d*x^3)*arctanh(x), and use the first 'd' terms in your series (above) to solve for the coefficients.

Looks like a simple enough expansion, but I don't know of any simple functions that expand in that exact form.