12.1.4What is a Taylor series?

A Taylor polynomial that has infinitely many terms is called a Taylor series. Taylor series that are centered at $x = 0$ are known as Maclaurin series.

In problem 12-16, you found that $p _ { 8 } ( x ) = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \frac { x ^ { 6 } } { 6 ! } + \frac { x ^ { 8 } } { 8 ! }$ represents the eighth-degree Taylor polynomial about $x = 0$ for $f(x) = \cos(x)$.

1. Use sigma notation to write an equation of the Maclaurin series for $f(x) = \cos(x)$. Notice that $p_8(x)$ is an eighth-degree polynomial, but it only has five terms.

2. Use graphing technology (or the 12-39 Student eTool (Desmos)) to complete the following tasks:  Click in the lower right corner of the graph to view it in full-screen mode.

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   Task 1: Compare the graphs of $y = p_2(x)$, $y = p_4(x)$, $y = p_6(x)$, and $y = p_8(x)$ to the graph of $f(x) = \cos(x)$. As the degree of $p$ increases, what happens to the relationship between $p$ and $f$?
   
   Task 2: As the number of terms of $p$ approaches infinity, make a prediction about the relationship between $p$ and  $f$.

The equation of a Taylor (or Maclaurin) series can be found by extending the equation of a Taylor polynomial. For example, in problem 12-18, you wrote the equation for the seventh-degree Taylor polynomial for $f(x) = e^x$ centered at $x = 0$ in two different ways:

Sigma form: $p _ { 7 } ( x ) = \displaystyle \sum _ { k = 0 } ^ { 7 } \frac { x ^ { k } } { k ! }$

Expanded form: $p _ { 7 } ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \frac { x ^ { 4 } } { 4 ! } + \frac { x ^ { 5 } } { 5 ! } + \frac { x ^ { 6 } } { 6 ! } + \frac { x ^ { 7 } } { 7 ! }$

1. Slightly alter the sigma form to represent this series to represent, $p(x)$, the Maclaurin series for $f(x) = e^x$.

2. The expanded form of the Maclaurin series for $f(x) = e^x$ must include a general term, which is the argument within the sigma notation. Alter the expanded form to represent an infinite series. Use “$+ … +$” before and “$+ …$ ” after the general term.

For $f(x) = e^x$, it is a celebrated fact that $\frac { d } { d x }(e^x) = e^x$ and $\int e ^ { x } d x= e^x + C$ .

1. What is the derivative of the Maclaurin series you found in 12-40, $p′(x)$. What do you notice?

2. Now integrate the Maclaurin series you found in 12-40, $\int p ( x ) d x$. What do you notice?

Refer back to problem 12-30, where you wrote a the fourth-degree Taylor polynomial about $x = 1$ for $f(x) = \ln(x)$. Write the first four terms and the general term of the Taylor series for $f$ centered at $x = 1$.

Set up an integral to represent the volume of the solid formed by semicircular cross-sections that are perpendicular to the $x$-axis with a base bounded by $y = e^{x}$ and $y = -2^{x}$ over the interval $[0, 4]$. Homework Help ✎

Calculate the area of the cardioid $r = 1 - \cos(θ )$ over the interval $[0, 2π]$. Homework Help ✎

Solve the differential equation $\frac { d y } { d x } = \frac { y ^ { 2 } - 1 } { x }$ if $y = 2$ when $x = 1$. Homework Help ✎

Integrate. Homework Help ✎

Determine the interval of convergence for each of the following series. Homework Help ✎