University of Florida/Egm4313/s12.team8.dupre/R2.7

Problem Statement
Develop the Maclaurin series (Taylor Series at t=0) for $$\displaystyle e^{t},cos(t),sin(t)$$ a) $$\displaystyle e^{t}$$    (7-1)  b)$$\displaystyle cos(t)$$    (7-2) c)$$\displaystyle sin(t)$$    (7-3)

Solution
For a Taylor series at any point t, the general form is as follows (to make it more clear, I have changed the usual t in (t-a) to a capital T): $$ \sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (T-a)^n=f(a) + \frac{f'(a)}{1!}(T - a) + \frac{f^2(a)}{2!}(T - a)^2 + \frac{f^3(a)}{3!}(T- a)^3 + \frac{f^4(a)}{4!}(T - a)^4 ... (7-4)$$

Part a solution
Using (7-4) as applied to (7-1) at time t=0 gives us the Maclaurin series for (7-1): $$\displaystyle e^t = \sum_{n=0}^{\infty} \frac{T^n}{n!}=e^0 + \frac{e^0}{1!}(T) + \frac{e^0}{2!}(T)^2 + \frac{e^0}{3!}(T)^3 + \frac{e^0}{4!}(T)^4 ... (7-6)$$ Simplifying this result, we obtain our final solution: $$\displaystyle e^t = 1 + T + \frac{T^2}{2} + \frac{T^3}{6} + \frac{T^4}{24} ... (7-7) $$

Part b solution
Using (7-4) as applied to (7-2) at time t=0 gives us the Maclaurin series for (7-2): $$\displaystyle cos(t) = \sum_{n=0}^{\infty} \frac{(-1)^nT^{2n}}{(2n)!} = cos(0) + \frac{-sin(0)}{1!}(T) + \frac{-cos(0)}{2!}(T)^2 + \frac{sin(0)}{3!}(T)^3 + \frac{cos(0)}{4!}(T)^4 + ... (7-8)$$ Simplifying this result, we obtain our final solution: $$\displaystyle cos(t) = \sum_{n=0}^{\infty} \frac{(-1)^nT^{2n}}{(2n)!} = 1-\frac{T^2}{4}+\frac{T^4}{24} ... (7-9)$$

Part c solution
Using (7-5) as applied to (7-3) at time t=0 gives us the Maclaurin series for (7-3): $$\displaystyle sin(t) = \sum_{n=0}^{\infty} \frac{(-1)^nT^{2n + 1}}{(2n + 1)!} = sin(0) + \frac{cos(0)}{1!}(T) + \frac{-sin(0)}{2!}(T)^2 + \frac{-cos(0)}{3!}(T)^3 + \frac{sin(0)}{4!}(T)^4 + ... (7-10)$$ Simplifying this result, we obtain our final solution: $$\displaystyle sin(t) = T - \frac{T^3}{6} + ... (7-11)$$