Chapter 8.3, Problem 25ES

Consider the initial-value problem $y^{\prime }=y,y\left(0\right)=1,$ and let $y_n$ denote the approximation of $y\left(1\right)$ using Euler's Method with n steps.

(a) What would you conjecture is the exact value of ${\lim }_{n\to +\infty }{y}_n?$ Explain your reasoning.

(b) Find an explicit formula for $y_n$ and use it to verify your conjecture in part (a).