I am studying Differential Geometry and came across this exercise
Let $c: I \rightarrow \mathbb{R}^n$ be a regular curve. Let $L(h)$ be the length of the curve between points $c(t_0)$ and $c(t_0 + h)$. Also define $d(h) := \lvert c(t_0 +h) - c(t_0) \rvert$.
Show that $$ \lim_{h\to 0} \frac{L(h) - d(h)}{(d(h))^3} = \frac{\kappa(t_0)^2}{24} $$ where $\kappa$ is the curvature of $c$.
I understand the geometric interpretation of the exercise and have shown that it is sufficient to show the statement for curves parameterized by arc length.
I also tried to use some kind of Taylor expansion of $c$ and $d$ but it did not work out.