Question

What is the Maclaurin series of `f(x) = cos(x)`?

Answer 1

A Maclaurin series is simply the Taylor series centered around `a = 0` specifically.

`sum_(n=1)^(N) (f^((n))(0))/(n!) x^n`
`= (f(0))/(0!)x^0 + (f'(0))/(1!)x^1 +(f''(0))/(2!)x^2 + (f'''(0))/(3!)x^3 + ...`

Thus, we need to take derivatives until we see a unique pattern.

`f^((0))(x) = color(green)(f(x) = cosx)`
`color(green)(f'(x) = -sinx)`
`color(green)(f''(x) = -cosx)`
`color(green)(f'''(x) = sinx)`
`color(green)(f''''(x) = cosx)`

So stopping at `n = 3` is fine. We can still figure out the rest by the pattern. We get:

`= (cos(0))/(0!)x^0 + cancel((-sin(0))/(1!)x^1)^(0) +(-cos(0))/(2!)x^2 + cancel((sin(0))/(3!)x^3)^(0) + ...`

`= 1/(0!) - x^2/(2!) + x^4/(4!) - x^6/(6!) + ...`

`color(blue)(= 1 - x^2/2 + x^4/(24) - x^6/(720) + ...)`