I'm trying to prove the error bound from the classical trapezoidal rule integral approximation, which states the error is $-(b-a)^3f''(c)/12n^2$ for some $c$ within the limits of integration.
In an attempt to prove it for a single trapezoid (and then sum up the errors to get an error for the whole integral), I tried averaging the two Taylor series expansions of $f$ at the two endpoints, and then integrating that, but the second term doesn't cancel out, so I'm left with an error based on the first derivative $f'$, not $f''$. Where am I going wrong?
I also can't find a proof of this online anywhere. It seems that everyone just states the fact and ignores the proof, or refers vaguely to costly textbooks, or (worse!) gives an incorrect proof.