The dilogarithm Li2(x) is a special function defined by the following integral
Expand the integrand in a power series and integrate term by term, thereby deriving the power series expansion for Li2 about x=0
f(x)= f(0) + xf'(0) + (x2/2)f''(0) + (x3/3!)f'''(0)
formula for Maclaurin series.
The Attempt at a Solution
So, as far as I can discern, I take the integrand of the dilogarithmic function, and treat it as a function itself by expanding it into the Taylor power series around x=0 (Maclaurin series). This means that I will be differentiating with respect to t, correct? Since I will later have to integrate term-by-term over t, so I treat the integrand as a function of t. The first two terms are pretty simple, but I have trouble getting past that. Here's what I have so far:
f(0) = ln(1-0*t)/t = ln(1) = 0
f'(0) = 1/(1-t)
I can't differentiate past this. Actually, I tried and came up with 1/2(1-t) which was nice because it followed a pattern, but when I integrated term by term, I ended up with natural logs of 0 which totally ruined the calculations.
I'm also not sure I understand the concept of expanding this sort of function into Taylor Series, the original function is of x, but the integral is over dt, do I expand using t as the variable or x? I know I'm suppose to approximate the function at x=0. Any insight on this problem would be great! I feel like I'm probably just differentiating poorly.