Here's the question:
Find a power series representation for the function and determine the
interval of convergence.
f(x) = 1/(x + 10)
well, $\displaystyle \frac x{2x^2 + 1} = \frac d{dx} \frac 14 \ln (2x^2 + 1) = \frac 14 \frac d{dx} \ln (1 - (-2x^2))$
and you know the power series for $\displaystyle \ln (1 - x)$, so just find it's derivative and plug it in...
EDIT: in retrospect, it is perhaps easier to think of this as x times the power series of 1/(1 + 2x^2) ...oh well