- #1
potentenum
- 1
- 0
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!