Activity 8.3.4

Consider the function $f(x) = \frac{1}{1-x}$. Find the degree $n$ Taylor polynomial approximation centered at $a = 0$ for $f$.

1. To begin finding $T_n(x)$, do the usual work of computing the various derivatives of $f$ and their respective values at $a = 0$; note that it's helpful to view $f(x)$ in the form $f(x) = (1-x)^{-1}$ so that we can easily compute the derivatives of $f$ using the chain rule. For instance, $$\begin{gathered} f'(x) = (-1)(1-x)^{-2}(-1) \end{gathered}$$ where the first “$-1$” arises from the power rule, while the second “$-1$” results from the chain rule, since $\frac{d}{dx}[1-x] = -1$. In order to see key patterns that arise, it's helpful not to combine the products of numbers that arise in the various derivatives. Record the first five derivatives of $f(x)$ in the spaces provided below. $$\begin{aligned} f(x) &= \frac{1}{1-x} = (1-x)^{-1} \\ f'(x) &= (-1)(1-x)^{-2}(-1) \\ f''(x) &= \underline{\hspace{2em}} \\ f'''(x) &= \underline{\hspace{2em}} \\ f^{(4)}(x) &= \underline{\hspace{2em}} \\ f^{(5)}(x) &= \underline{\hspace{2em}} \end{aligned}$$

2. Next, evaluate the derivatives you determined in (a) at $a = 0$ and use these to find the values of the coefficients of the Taylor polynomial centered at $a = 0$. Record your work in the blank spaces provided. $$\begin{aligned} k &= 0 & f(0) &= \frac{1}{1-0} = 1 & c_0 &= f(0) = 1 \\ k &= 1 & f'(0) &= \frac{(-1)}{(1-0)^{2}} \cdot (-1) = 1 & c_1 &= \frac{f'(0)}{1!} = \frac{1}{1!} = 1 \\ k &= \underline{\hspace{2em}} & f''(0) &= \underline{\hspace{2em}} & c_2 &= \frac{f''(0)}{2!} = \underline{\hspace{2em}} \\ k &= \underline{\hspace{2em}} & f'''(0) &= \underline{\hspace{2em}} & c_3 &= \underline{\hspace{2em}} \\ k &= \underline{\hspace{2em}} & f^{(4)}(0) &= \underline{\hspace{2em}} & c_4 &= \underline{\hspace{2em}} \\ k &= \underline{\hspace{2em}} & f^{(5)}(0) &= \underline{\hspace{2em}} & c_5 &= \underline{\hspace{2em}} \end{aligned}$$

3. What pattern do you observe in the value of $c_k$? State the degree $5$ Taylor polynomial, $T_5(x)$, as well as the formula you expect for the general degree $n$ Taylor polynomial, $T_n(x)$.

4. By identifying the value of $r$, explain why the degree $n$ Taylor polynomial $T_n(x)$ can be thought of as a finite geometric series.

5. What is the Taylor series centered at $a = 0$ for $f(x) = \frac{1}{1-x}$? (As we will see in the next section, the “Taylor series” of a function is the infinite series that results by simply extending the Taylor polynomials indefinitely.)