A student asked me for help on the following problem:
Suppose that the bending moment $M(x)$ can be modeled via the vertical displacement $w(x)$ of a beam where $E$ and $I$ are constant. $$ \frac{M(x)}{EI} = -\frac{w''(x)}{[1 + w'(x)^2]^\frac{3}{2}} $$ Write $\frac{M(x)}{EI}$ as a power series in terms of $w'$.
My first thought was to write $$ -\frac{w''(x)}{[1 + w'(x)^2]^\frac{3}{2}} = \frac{d}{dx}\left[-\frac{w'(x)}{\sqrt{1 + w'(x)^2}}\right], $$ but I'm not sure how to write this as a power series either.
I worked with the student and we were able to figure it out.
Simply factor and use the binomial series: $$ -w''(x)\frac{1}{[1 + w'(x)^2]^\frac{3}{2}} = -w''(x)\sum_{n=0}^\infty \binom{-3/2}{n} w'(x)^{2n}. $$