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I tried doing the difference between the taylor series expansions of $f(x+h)$ and $f(x-h)$, but it didn't reflect the answer that I should get using the third order central divided difference. The approximation for the third derivative using a central difference is $$\dfrac{f(x+2h)-2[f(x+h)-f(x-h)]-f(x-2h)}{2h^3}$$ Should I try the expansion of $f(x+2h)+f(x+h)-(f(x)+f(x-h))$?

John
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  • @IvanNeretin How do you know which expression to use? And what do you mean by 3 and 1. Is 3 = f(x+2h) and 1= f(x-h) – John Dec 15 '15 at 12:04

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There exist different Taylor expansions but to prove the one you have, you can try this:

  • Write the third order expansion of f at x-h and x+h.
  • Write the third order expansion of f at x-2h and x+2h.
  • Write the difference D1 of the expansions at x-h and x+h
  • Write the difference D2 of the expansions at x-2h and x+2h

Here D1 and D2 will look like:

$$ D_1 = hf'+\frac{1}{3}h^3f'''$$ $$ D_2 = 2hf'+\frac{8}{3}h^3f'''$$

To get rid of the $f'$, write $$D_2-2D_1 = 2h^3f'''$$

abdeaitali
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