# Closed form expression for f(x) = sigma (n = 1 to infinity) for x^n / [n(n+1)]

## Homework Statement

Consider the power series.

sigma (n=1 to infinity) x^n / [n(n+1)]

if f(x) = sigma x^n / [n(n+1)], then compute a closed-form expression for f(x).

It says: "Hint: let g(x) = x * f(x) and compute g''(x). Integrate this twice to get back to g(x) and hence derive f(x)".

## Homework Equations

None? Well, see number 3.

## The Attempt at a Solution

OK so i calculated the interval of convergence to be

-1 <= x <=1

and i'm following the hint so.

g(x) = x f( x)
and f(x) = x/2 + x²/6 + x^3 / 12 + x^4 / 20 ...

and x f(x) = x²/2 + x^3 /6 + x^4 / 12 ...

and g(x) = above.

so g'(x) = x + x²/2 + x^3 / 3 + x^4 / 4...

and g''(x) = 1 + x + x² + x^3 + x^4

and I recognized g''(x) to be the power series at x = 0 for

1 / (1 - x)

so i set g'' (x) = 1 / (1-x)

and then the hint said to integrate g''(x) twice, so

g'(x) = - ln (1-x).

now i'm stuck. How can I integrate -ln (1-x) again? or am i supposed to use what i know (i.e. f(1) = 1 and f(-1) = -1 - ln2?) (that is, i calculated the sum of f(x) if x = 1 and x = -1.

THANK YOU!!!