Prove that the “rate of convergence” for Euler’s method in Problem 12 is comparable to 1...

Prove that the “rate of convergence” for Euler’s method in Problem 12 is comparable to 1/n by showing that

[Hint: Use L’Hôpital’s rule and the Maclaurin expansion for In (1 + t).]

12. - In Example 2 we approximated the transcendental number e by using Euler’s method to solve the initial value problem

Show that the Euler approximation yn obtained by using the step size 1/n is given by the formula

Recall from calculus that

and hence Euler’s method converges (theoretically) to the correct value.