Activity 8.5.3.
In Activity 8.5.2 we determined small order Taylor polynomials for a few familiar functions, and also found general patterns in the derivatives evaluated at \(0\text{.}\) Use that information to write the Taylor series centered at \(0\) for the following functions.
(a)
\(f(x) = \frac{1}{1-x}\)
(b)
\(f(x) = \cos(x)\) (You will need to carefully consider how to indicate that many of the coefficients are 0. Think about a general way to represent an even integer.)
(c)
\(f(x) = \sin(x)\) (You will need to carefully consider how to indicate that many of the coefficients are \(0\text{.}\) Think about a general way to represent an odd integer.)
(d)
\(f(x) = \frac{1}{1+x}\)
