how do you prove this? I couldnt do it
show that if f has 4 continuous derivatives then
(g(i+1) -2g(i) + g(i-1)/h^2 = g''(i) + O(h^2)
You say that $\displaystyle f$ has 4 continuous derivatives but then ask about g?
Also, does the problem really say "4 continuous derivatives"? This is true of any function that has 2 continuous functions. Also can we take it that "i" is just a variable and not such that $\displaystyle i^2= -1$?
Finally, where did that "h" come from? Are you sure the final equation shouldn't be (f(i+h)- 2f(i)+ f(i- h))/h^2= f''(i)+ O(h^2) ??
Write f(x) as a Taylor's polynomial of degree 2, about x= i, and do the algebra to find the left side of that equation.