Activity 8.3.4.
Consider the function \(f(x) = \frac{1}{1-x}\text{.}\) Find the degree \(n\) Taylor polynomial approximation centered at \(a = 0\) for \(f\text{.}\)
(a)
To begin finding \(T_n(x)\text{,}\) do the usual work of computing the various derivatives of \(f\) and their respective values at \(a = 0\text{;}\) 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{equation*}
f'(x) = (-1)(1-x)^{-2}(-1)
\end{equation*}
where the first “\(-1\)” arises from the power rule, while the second “\(-1\)” results from the chain rule, since \(\frac{d}{dx}[1-x] = -1\text{.}\) 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{align*}
f(x) \amp= \frac{1}{1-x} = (1-x)^{-1}\\
f'(x) \amp= (-1)(1-x)^{-2}(-1)\\
f''(x) \amp= \fillinmath{XXXXXXXXXX} \\
f'''(x) \amp= \fillinmath{XXXXXXXXXX} \\
f^{(4)}(x) \amp= \fillinmath{XXXXXXXXXX} \\
f^{(5)}(x) \amp= \fillinmath{XXXXXXXXXX}
\end{align*}
(b)
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 \(a = 0\text{.}\) Record your work in the blank spaces provided.
\begin{align*}
k \amp= 0 \amp f(0) \amp= \frac{1}{1-0} = 1 \amp c_0 \amp= f(0) = 1\\
k \amp= 1 \amp f'(0) \amp= \frac{(-1)}{(1-0)^{2}} \cdot (-1) = 1 \amp c_1 \amp= \frac{f'(0)}{1!} = \frac{1}{1!} = 1\\
k \amp= \fillinmath{X} \amp f''(0) \amp= \fillinmath{XXXXX} \amp c_2 \amp= \frac{f''(0)}{2!} = \fillinmath{XXXX}\\
k \amp= \fillinmath{X} \amp f'''(0) \amp= \fillinmath{XXXXX} \amp c_3 \amp= \fillinmath{XXXXXXXXX}\\
k \amp= \fillinmath{X} \amp f^{(4)}(0) \amp= \fillinmath{XXXXX} \amp c_4 \amp= \fillinmath{XXXXXXXXX} \\
k \amp= \fillinmath{X} \amp f^{(5)}(0) \amp= \fillinmath{XXXXX} \amp c_5 \amp= \fillinmath{XXXXXXXXX}
\end{align*}
(c)
What pattern do you observe in the value of \(c_k\text{?}\) State the degree \(5\) Taylor polynomial, \(T_5(x)\text{,}\) as well as the formula you expect for the general degree \(n\) Taylor polynomial, \(T_n(x)\text{.}\)
(d)
By identifying the value of \(r\text{,}\) explain why the degree \(n\) Taylor polynomial \(T_n(x)\) can be thought of as a finite geometric series.
(e)
What is the Taylor series centered at \(a = 0\) for \(f(x) = \frac{1}{1-x}\text{?}\) (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.)
