# Taylor Series of a Pair of Functions

• February 9th 2010, 08:42 AM
LithiumPython
Taylor Series of a Pair of Functions
I have a problem that asks for the Taylor series of the following functions:

$f(x)=sin(a)cos(x)+cos(a)sin(x)$
$g(x)=sin(a+x)$
In both cases a is a constant

Knowing that both of these represent the formula for addition of sines, they both yield the same Taylor Series

$sin(a)+cos(a)x-\frac{1}{2}cos(a)(x^2)-\frac{1}{6}cos(a)(x^3)+\frac{1}{24}sin(a)(x^4)+...$

My question regards the form of the nth term of the Taylor series, would:

$(sin(a)) \frac{(-1)^nx^{2n}}{(2n)!} + (cos(a)) \frac{(-1)^nx^{2n+1}}{(2n+1)!}$

be a valid way to express the nth term of the series?