It is easy to see that Euler's method converges for the special case of the equation with solution . For this example,
Error estimates show
The error analysis above ignores round-off error. If one assumes that a fixed error is added at each time step, then the error estimate of is modified to . That is, taking more steps reduces the discretization error, but increases the round-off error. Therefore, there is a point of diminishing returns where the total error increases as h decreases. Better results require not more effort, but more efficiency. The key is to take more terms of the Taylor series and reduce the discretization error to O(hp), with p>1.