Thread: Convergence of Eulers method.. Differential equations

Prove that the approximation of Euler's method converges to the exact solution at any fixed point as h->0.
- Because h->0 we know that the number of intervals h approaches infinity. I just can't seem to come up with a way of proving this theorem using algebra.

Originally Posted by mathlete2

Prove that the approximation of Euler's method converges to the exact solution at any fixed point as h->0.
- Because h->0 we know that the number of intervals h approaches infinity. I just can't seem to come up with a way of proving this theorem using algebra.

Look at the error analysis, the error per step should be bounded above by , for some (depending on the function and the interval over which we are integrating) and the number of steps will be where is the length of the interval we are integrating over. So the total error at a fixed point will be bounded by , and so will go to zero as goes to zero.

RonL