R Notebook

Taylor Series

\[ \cos{x} = \sum_{n=\infty}(-1)^n \frac{x^{2n}}{(2n)!} \] without loss of generality, we can look at the case where x = 1, such that.

\[ \cos{(1)} = \sum_{n=\infty}(-1)^n \frac{1^{2n}}{(2n)!} \\ = \sum_{n=\infty}(-1) \frac{1}{(2n)!} \] Then, suppose that x = -1, such that

\[ \cos{(-1)} = \sum_{n=\infty}(-1)^n \frac{-1^{2n}}{(2n)!} \\ = \sum_{n=\infty}(-1) \frac{1}{(2n)!} \] Because we’re raising the x term to an even exponent, the sign of the exponent of the base number doesn’t matter. Notice how this is not the case of the Taylor Series for the \(\sin(x)\) function.

\[ \sin(x) = \sum_{n=\infty}(-1)^n \frac{x^{2n+1}}{2n+1} \] Notice how the top term in the fraction has an odd exponent. This ensures that sign matters. I believe this is at least part of the etymology behind the terms ‘even’ and ‘odd’ functions.