Exercise 8

The examples we have considered so far in this section have all been for Taylor polynomials and series centered at 0, but Taylor polynomials and series can be centered at any value of $a$.

1. Let $f(x) = \cos(x)$. Find the Taylor polynomials up through order four of $f$ centered at $a = \frac{\pi}{2}$. Then find the Taylor series for $f(x)$ centered at $a = \frac{\pi}{2}$. Why is the result not surprising?

2. Let $f(x) = \frac{1}{1+x} = (1+x)^{-1}$. Find the Taylor polynomials up through order four of $f$ centered at $a = 1$. Then find the Taylor series for $f(x)$ centered at $a = 1$.