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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.

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Is this OK? Here is a link to the first page of a proof in Mathematics Magazine. There is also this video on YouTube. If you type

trapezoid rule error proof

into Google, you get these, and more.

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    $\begingroup$ Is this not possible with Taylor series approximations and Lagrange errors? That seemed like the most natural way to prove it. $\endgroup$ – JeremyKun Dec 15 '11 at 23:22
  • $\begingroup$ You asked for a proof online. I found you three, with a method for finding more. Now you have changed your mind, and you want a proof using Taylor and Lagrange. So I suggest looking at the proofs I have pointed to until you either find one that uses Taylor and Lagrange or until the absence of such proofs convinces you it's impossible (or encourages you to have another go at the problem yourself). Or maybe type trapezoid rule error proof Taylor Lagrange into Google. $\endgroup$ – Gerry Myerson Dec 15 '11 at 23:49
  • $\begingroup$ @Bean: You'll want to look up Euler-Maclaurin. $\endgroup$ – J. M. is a poor mathematician Dec 16 '11 at 1:58

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