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

Problem Statement

Develop the Maclaurin series (Taylor Series at t=0) for $e^t,cos(t),sin(t)$

a) $e^t$ (7-1)

b)$cos(t)$ (7-2)

c)$sin(t)$ (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):

$\sum _{n=0}^{\mathrm{\infty }}\frac{f^n(a)}{n!}(T-a{)}^n=f(a)+\frac{f^{\prime }(a)}{1!}(T-a)+\frac{f^2(a)}{2!}(T-a{)}^2+\frac{f^3(a)}{3!}(T-a{)}^3+\frac{f^4(a)}{4!}(T-a{)}^4...(7-4)$

Part a solution

Using (7-4) as applied to (7-1) at time t=0 gives us the Maclaurin series for (7-1):

$e^t=\sum _{n=0}^{\mathrm{\infty }}\frac{T^n}{n!}=e^0+\frac{e^0}{1!}(T)+\frac{e^0}{2!}(T{)}^2+\frac{e^0}{3!}(T{)}^3+\frac{e^0}{4!}(T{)}^4...(7-6)$

Simplifying this result, we obtain our final solution:

$e^t=1+T+\frac{T^2}{2}+\frac{T^3}{6}+\frac{T^4}{24}...(7-7)$

Part b solution

Using (7-4) as applied to (7-2) at time t=0 gives us the Maclaurin series for (7-2):

$cos(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{T}^{2n}}{(2n)!}=cos(0)+\frac{-sin(0)}{1!}(T)+\frac{-cos(0)}{2!}(T{)}^2+\frac{sin(0)}{3!}(T{)}^3+\frac{cos(0)}{4!}(T{)}^4+...(7-8)$

Simplifying this result, we obtain our final solution:

$cos(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{T}^{2n}}{(2n)!}=1-\frac{T^2}{4}+\frac{T^4}{24}...(7-9)$

Part c solution

Using (7-5) as applied to (7-3) at time t=0 gives us the Maclaurin series for (7-3):

$sin(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{T}^{2n+1}}{(2n+1)!}=sin(0)+\frac{cos(0)}{1!}(T)+\frac{-sin(0)}{2!}(T{)}^2+\frac{-cos(0)}{3!}(T{)}^3+\frac{sin(0)}{4!}(T{)}^4+...(7-10)$

Simplifying this result, we obtain our final solution:

$sin(t)=T-\frac{T^3}{6}+...(7-11)$