1. How geometric series naturally connect to Taylor polynomials

How geometric series naturally connect to Taylor polynomials

2. Summary

Summary

3. To begin finding $T_n(x)$

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,

f'(x) = (-1)(1-x)^{-2}(-1)

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) &= \\ f'''(x) &= \\ f^{(4)}(x) &= \\ f^{(5)}(x) &= \end{aligned}

4. Next

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 &= & f''(0) &= & c_2 &= \frac{f''(0)}{2!} = \\ k &= & f'''(0) &= & c_3 &= \\ k &= & f^{(4)}(0) &= & c_4 &= \\ k &= & f^{(5)}(0) &= & c_5 &= \end{aligned}

5. What pattern do you …

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, $T_n(x)$ .

6. By identifying the v…

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.

7. What is the Taylor s…

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.)

8. In Equation

In Equation, we learned that for an infinite geometric series

S = a + ar + ar^2 + \cdots + ar^{n-1} + \cdots

, if $|r| \lt 1$ , then

S = \frac{a}{1-r}

. This result provides a quick way to determine the value of an infinite geometric series that converges.

In this exercise, we explore why the infinite geometric series fails to converge when $r = 1$ , $r = -1$ , and $|r|>1$ .

If we consider the finite geometric series that results when $a = 1$ and $r = 1$ , we get the sum

S_n = 1 + 1 + \cdots + 1

, where there are $n$ terms in the sum. - Compute $S_2$ , $S_3$ , and $S_4$ . What is the general formula for $S_n$ ? - Explain why the sequence $S_n$ does not converge to a finite value. Contrast this with, for example, our work in Example 8.3.4. - Next, consider the finite geometric series that results when $a = 1$ and $r = -1$ , which is the sum

S_n = 1 - 1 + 1 - 1 + \cdots + (-1)^{n-1}

, where there are $n$ terms in the sum. (We view the first “ $1$ ” as resulting from “ $1 \cdot (-1)^0$ ”.) - Compute $S_2$ , $S_3$ , $S_4$ , and $S_5$ . What do you observe? - Explain why the sequence $S_n$ does not converge to a finite value. Contrast this with, for example, our work in Example 8.3.4. - Now consider the geometric sum with $a = 1$ and $r = 2$ , so

S_n = 1 + 2 + 4 + \cdots + 2^{n-1}

. - Compute $S_2$ , $S_3$ , and $S_4$ . What is the general formula for $S_n$ ? - What do you observe happens to these partial sums as $n$ increases without bound? What does this tell us about the infinite geometric series $1 + 2 + 4 + \cdots + 2^{n-1} + \cdots$ ?

9. Suppose you drop a g…

Suppose you drop a golf ball onto a hard surface from a height $h$ . The collision with the ground causes the ball to lose energy and so it will not bounce back to its original height. The ball will then fall again to the ground, bounce back up, and continue. Assume that at each bounce the ball rises back to a height $\frac{3}{4}$ of the height from which it dropped. Let $h_n$ be the height of the ball on the $n$ th bounce, with $h_0 = h$ . In this exercise we will determine the distance traveled by the ball and the time it takes to travel that distance. - Determine a formula for $h_1$ in terms of $h$ . - Determine a formula for $h_2$ in terms of $h$ . - Determine a formula for $h_3$ in terms of $h$ . - Determine a formula for $h_n$ in terms of $h$ . - Write an infinite series that represents the total distance traveled by the ball. Then determine the value of this series. - Next, let's determine the total amount of time the ball is in the air. - When the ball is dropped from a height $H$ , if we assume the only force acting on it is the acceleration due to gravity, then the height of the ball at time $t$ is given by

H - \frac{1}{2}gt^2

. Use this formula to determine the time it takes for the ball to hit the ground after being dropped from height $H$ . - Use your work in the preceding item, along with that in (a)-(e) above to determine the total amount of time the ball is in the air.

10. It is important to u…

It is important to understand the power of geometric growth compared to linear growth. Suppose you are hired for a job that will take you 30 days to complete and are offered two options for how you'll be compensated.

Option 1. You can be paid ＄500 per day, or Option 2. You can be paid 1 cent the first day, 2 cents the second day, 4 cents the third day, 8 cents the fourth day, and so on, doubling the amount you are paid each day.

How much will you be paid for the job in total under Option 1? - Complete Preview Activity 8.2 to determine the pay you will receive under Option 2 for the first 10 days. - Find a formula for the amount paid on day $n$ , as well as for the total amount paid by day $n$ . Use this formula to determine which option (1 or 2) you should take.