Find a power series representation for the function f(x)=ln(1-x^2). Write the series in sigma notation. For which values of x is the power series representation of f valid?
for $\displaystyle |x|<1$ ...
$\displaystyle \frac{1}{1-x} = 1 + x + x^2 + x^3 + ...$
integrate ...
$\displaystyle -\ln(1-x) = C + x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ...
$
for $\displaystyle x = 0$ , $\displaystyle C = 0$ ...
$\displaystyle \ln(1-x) = -\left(x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ... \right)$
$\displaystyle \ln(1-x^2) = -\left(x^2 + \frac{x^4}{2} + \frac{x^6}{3} + \frac{x^8}{4} + ... \right)$
I'll leave you to find the sigma notation for the series.