# 4.7: Numerical Experiment (Approximating e^jθ)

We have demonstrated that the function $$e^{jθ}$$ has two representations:

1. $$e^{jθ}=\lim_{n→∞}(1+\frac {jθ} n)^n$$  ; and
2. $$e^{jθ}=\lim_{n→∞}∑_{k=0}^n\frac {(jθ)^k} {k!}$$

In this experiment, you will write a MATLAB program to evaluate the two functions $$f_n$$ and $$S_n$$ for twenty values of n:

1. $$f_n=(1+\frac {jθ} n)^n,\;n=1,2,...,20$$ ; and
2. $$S_n=∑^n_{k=0}\frac {(jθ)^k} {k!},\;n=1,2,...,20k$$

Choose $$θ=π/4(=\mathrm{pi}/4)$$. Use an implicit for loop to draw and plot a circle of radius 1. Then use an implicit for loop to compute and plot $$f_n$$ and an explicit for loop to compute and plot $$S_n$$ for n=1,2,...,100. You should observe plots like those illustrated in the Figure. Interpret them.