The final answer has the form:

Fill in the following table to find the coefficients:

The first four terms of the Taylor Series for the function centered at are

We find the Taylor Series for a function.

In this section, we will find a power series expansion centered at for a given infinitely differentiable function, . Such a series is called a Taylor Series. In the case that , we call the series a Maclaurin Series.

In other words, for a given function and a given center, , we wish to create a power series so that where the series converges in an interval of the form for some . This is an interval centered at of radius . The key to creating the Taylor Series is to find the coefficients, . To find , let in the equation above. We get: Next, to find , we differentiate both sides of the equation above, and then let . Differentiating gives and substituting in , this simplifies to

Next, to find , we differentiate both sides of the equation above, and then let . Differentiating gives and substituting in , this simplifies to so that .

Continuing in this fashion, we can find : and letting gives so that . The pattern continues, so that and . In general, we have Thus, the Taylor Series representation of centered at is given by

We present this results in the following theorem.

Taylor Series Suppose the function has derivatives of all orders at . Then the **Taylor
Series** representation for centered at is given by

In the case that , the series has the form and is called the **Maclaurin Series** for
.

example 1 Find the first four terms of the Taylor Series centered at .

Since we are asked to find the first four terms, we must find the coefficients and and the answer will have the form To find the coefficients, we create the following table:

Since we are asked to find the first four terms, we must find the coefficients and and the answer will have the form To find the coefficients, we create the following table:

We can now use the coefficients in the right-hand column of the table to write the final answer. The first four terms of the Taylor Series for the function centered at are \begin{align*} \sqrt x &= c_0 + c_1(x-4) + c_2(x-4)^2 + c_3(x-4)^3 + \cdots \\ &= 2 + \frac 14(x-4) -\frac{1}{64}(x-4)^2 + \frac{1}{512}(x-4)^3 + \cdots \end{align*}

(problem 1) Find the first four terms of the Taylor Series centered at .

The final answer has the form:

a) b) c) d)

Fill in the following table to find the coefficients:

The first four terms of the Taylor Series for the function centered at are

In the last example and problem, we computed only the first four terms of the Taylor Series, yielding a polynomial of degree 3. This is a special polynomial.

Taylor Polynomial Give a function and its first derivatives at , , the polynomial is
called the Taylor Polynomial of degree for centered at .

The Taylor Polynomial of degree 1 for centered at , is also known as the
Linearization of at which is used in making linear approximations.

example 2 Find the Taylor Polynomial of degree 3, , for centered at .

Since we are asked to find , we must find the coefficients and and the answer will have the form To find the coefficients, we create the following table:

Since we are asked to find , we must find the coefficients and and the answer will have the form To find the coefficients, we create the following table:

We can now use the coefficients in the right-hand column of the table to write the final answer. The Taylor Polynomial of degree 3 for centered at is \begin{align*} T_3(x) &= c_0 + c_1(x+1) + c_2(x+1)^2 + c_3(x+1)^3 \\ &= -1 + \frac 13(x+1) + \frac{1}{9}(x+1)^2 + \frac{5}{81}(x+1)^3 \end{align*}

(problem 2a) Find the Taylor Polynomial of degree 3, , for centered at .

The final answer has the form:

a) b) c) d)

Fill in the following table to find the coefficients:

The first four terms of the Taylor Series for the function centered at are

(problem 2b) Find the Taylor Polynomial of degree 3, , for centered at .

The function can be written as a power function, , for an appropriate value of . Write as a power function:

The final answer has the form:

a) b) c) d)

Fill in the following table to find the coefficients:

The first four terms of the Taylor Series for the function centered at are

example 3 Find the Taylor Polynomial of degree 4, , for centered at .

Since we are asked to find the forth degree Taylor Polynomial, we must find the coefficients and and the answer will have the form To find the coefficients, we create the following table:

Since we are asked to find the forth degree Taylor Polynomial, we must find the coefficients and and the answer will have the form To find the coefficients, we create the following table:

We can now use the coefficients in the right-hand column of the table to write the final answer. The first five terms of the Taylor Series for the function centered at are \begin{align*} T_4(x) &= c_0 + c_1(x-1) + c_2(x-1)^2 + c_3(x-1)^3 + c_4(x-1)^4\\ &= 0 + 1(x-1) -\frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3-\frac{1}{4}(x-1)^4\\ &= (x-1) -\frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3-\frac{1}{4}(x-1)^4 \end{align*}

(problem 3) Find the Taylor Polynomial of degree 4, , for centered at .

The final answer has the form:

a) b) c) d)

Fill in the following table to find the coefficients:

The first five terms of the Taylor Series for the function centered at are

Taylor Polynomials can be used to approximate function values, and the following theorem gives us an expression for the error in such an approximation.

Taylor’s Remainder Theorem If the function is times differentiable at then
the difference between and is given by where is number between and
.

Here is a detailed, lecture style video on Taylor Series:

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Here is a detailed, lecture style video on Maclaurin Series:

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