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10.15 Representing Functions as Power Series

3 min readjune 7, 2020

Congrats congrats! You made it to the last AP Calculus BC topic! In this key topic, you’ll be deriving power series using different techniques you’ve learned throughout this course.

🟦Power Series

A power series, similar to what you’ve learned about Taylor polynomials, are a representation of a function using an infinite series of polynomials. It is generally expressed by the following below, where n is a non-negative integer, ana_n is a sequence of real numbers, and r is a real number.

n=0an(xr)\displaystyle\sum_{n=0}^{∞}{a_n(x-r)}

For the AP Calculus BC exam, memorizing these three frequently appearing power series can be a lifesaver: exe^x, sin(x)\sin(x), and cos(x)\cos(x). In a lot of cases, you will be able to use the original series to find the power series of a transformed version of one of these functions.

ex=n=0xnn!=1+x+x22!+x33!++xnn!e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots+\frac{x^n}{n!}
cos(x)=n=0(1)nx2n(2n)!=1x22!+x44!x66!++(1)nx2n(2n)!\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots+\frac{(-1)^nx^{2n}}{(2n)!}
sin(x)=n=0(1)nx2n+1(2n+1)!=xx33!+x55!x77!++(1)nx2n+1(2n+1)!\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots+\frac{(-1)^nx^{2n+1}}{(2n+1)!}

✏️ Power Series Example 1

Find the power series representation for x2exx^2e^x. Include the first 4 nonzero terms and the general term.

We know that the power series of exe^x is 1+x+x22!+x33!+...+xnn!+...1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+.... So, since the function is just exe^x being multiplied by x2x^2, we can just multiply the power series of exe^x by x2x^2:

x2ex=x2[1+x+x22!+x33!+...+xnn!+...]x^2e^x=x^2[1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+...]
=x2+x3+x42!+x53!+...+xn+2n!+...=x^2+x^3+\frac{x^4}{2!}+\frac{x^5}{3!}+...+\frac{x^{n+2}}{n!}+...

Great job! Lets look at one more example 🤗

✏️ Power Series Example 2

If h(x)h(x) is the power series centered at x=0x=0 of cos(x)\cos(x), what is h(x)h’(x)? Include the the first 3 nonzero terms and the general term.

We’re given that h(x)=1x22!+x44!+...+(1)nx2n(2n)!+...h(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...+\frac{(-1)^nx^{2n}}{(2n)!}+...To find h(x)h’(x), simply have to take the derivative of the series:

h(x)=x+x33!x55!+...+(1)nx2n1(2n1)!+...h'(x)=-x+\frac{x^3}{3!}-\frac{x^5}{5!}+...+\frac{(-1)^nx^{2n-1}}{(2n-1)!}+...

A cool thing to recognize is that this series is equivalent to the power series of sin(x)\sin(x), but negative; this proves that the derivative of cos(x)\cos(x) is sin(x)-\sin(x).


🔷Practice FRQ 2022 #6

The following free-response question (FRQ) is Question 6 from the 2022 AP Calculus BC examination administered by College Board. All credit to College Board.

The function f is defined by the power series f(x)=xx33+x55x77+...+(1)nx2n+12n+1+...f(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+...+\frac{(-1)^nx^{2n+1}}{2n+1}+... for all real numbers x for which the series converges.

c) Write the first four nonzero terms and the general term for an infinite series that represents f′(x).

For this problem, all you have to do is take the derivative, similar to our example 2:

f(x)=1x2+x4x6+...+(1)nx2n+...f'(x)=1-x^2+x^4-x^6+...+(-1)^nx^{2n}+...

Awesome work! 🎉


👏 Congrats! You’re done.

Congratulations, you’re done with this unit, and as a result, you’ve also reached the end of AP Calculus BC! Now, you have all the tools you need to ace that AP test this May!

As you start to review, be sure to check out our Study Tools unit for more resources and exam information.

Key Terms to Review (20)

Complex Analysis

: Complex analysis is a branch of mathematics that focuses on studying functions involving complex numbers. It explores concepts like holomorphic functions, contour integrals, and residues.

Complex Numbers

: Complex numbers consist of both real and imaginary parts represented by a+bi form (where "a" is the real part and "bi" is the imaginary part). They extend our number system beyond just real numbers.

Differential Equations

: Differential equations are mathematical equations that involve derivatives. They describe how a function changes over time or in relation to other variables.

Epsilon-Delta Definition of a Limit

: The epsilon-delta definition is a rigorous way to define limits in calculus. It states that for every positive value of epsilon (ε), there exists a positive value of delta (δ) such that if the distance between x and c is less than delta, then the distance between f(x) and L is less than epsilon.

Fields

: Fields are mathematical structures that satisfy certain properties, allowing for addition, subtraction, multiplication, and division. They are sets with two operations (addition and multiplication) that follow specific rules.

Fundamental Theorem of Calculus

: The Fundamental Theorem of Calculus states that if f(x) is continuous on an interval [a, b] and F(x) is any antiderivative of f(x), then ∫[a,b] f(x) dx = F(b) - F(a). In simpler terms, it relates differentiation and integration by showing that finding the area under a curve can be done by evaluating its antiderivative at two points.

Geometry

: Geometry is the branch of mathematics that studies shapes, sizes, positions, angles, and dimensions of objects in space. It explores properties such as symmetry, congruence, similarity, and transformations.

Hyperbolic Functions

: Hyperbolic functions are a set of mathematical functions that are analogs to the trigonometric functions. They are defined in terms of exponential functions and can be used to model various physical phenomena.

Linear Algebra

: Linear algebra is a branch of mathematics that deals with vector spaces and linear transformations. It involves studying systems of linear equations, matrices, and their properties.

Maclaurin Polynomial

: A Maclaurin polynomial is a type of Taylor polynomial that is centered at the point x = 0. It is used to approximate a function by adding up terms of different powers of x.

Maclaurin Series

: A Maclaurin series is a special case of Taylor series expansion, where the expansion is centered around x = 0 (or when h = 0).

Multivariable Calculus

: Multivariable calculus is a branch of calculus that deals with functions of multiple variables, where the variables can change simultaneously. It involves studying concepts such as partial derivatives, multiple integrals, and vector fields.

Number Theory

: Number theory is the branch of mathematics that deals with properties and relationships of numbers, particularly integers. It focuses on studying patterns, divisibility, prime numbers, and other number-related concepts.

Power Series

: A power series is an infinite series that represents a function as an infinite polynomial expression.

Reaction Rates

: Reaction rates refer to the speed at which a chemical reaction takes place. It measures how quickly reactants are consumed or products are formed.

Real Analysis

: Real analysis is a branch of mathematics that deals with the study of real numbers and their properties, including limits, continuity, differentiation, and integration.

Taylor Series

: A Taylor series is an expansion of a function into an infinite sum of terms, where each term represents the contribution from different derivatives of the function at a specific point.

Topology

: Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations, such as stretching or bending. It focuses on the concept of "closeness" and the relationships between points, without considering specific distances or measurements.

Trigonometric Substitutions

: Trigonometric substitutions are techniques used to simplify integrals involving radical expressions or quadratic equations. By substituting trigonometric functions for variables, these integrals can be transformed into simpler forms that are easier to solve.

Vector Calculus

: Vector calculus is a branch of mathematics that focuses on studying vector fields and their properties. It involves operations such as differentiation and integration applied to vectors and vector-valued functions.

10.15 Representing Functions as Power Series

3 min readjune 7, 2020

Congrats congrats! You made it to the last AP Calculus BC topic! In this key topic, you’ll be deriving power series using different techniques you’ve learned throughout this course.

🟦Power Series

A power series, similar to what you’ve learned about Taylor polynomials, are a representation of a function using an infinite series of polynomials. It is generally expressed by the following below, where n is a non-negative integer, ana_n is a sequence of real numbers, and r is a real number.

n=0an(xr)\displaystyle\sum_{n=0}^{∞}{a_n(x-r)}

For the AP Calculus BC exam, memorizing these three frequently appearing power series can be a lifesaver: exe^x, sin(x)\sin(x), and cos(x)\cos(x). In a lot of cases, you will be able to use the original series to find the power series of a transformed version of one of these functions.

ex=n=0xnn!=1+x+x22!+x33!++xnn!e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots+\frac{x^n}{n!}
cos(x)=n=0(1)nx2n(2n)!=1x22!+x44!x66!++(1)nx2n(2n)!\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots+\frac{(-1)^nx^{2n}}{(2n)!}
sin(x)=n=0(1)nx2n+1(2n+1)!=xx33!+x55!x77!++(1)nx2n+1(2n+1)!\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots+\frac{(-1)^nx^{2n+1}}{(2n+1)!}

✏️ Power Series Example 1

Find the power series representation for x2exx^2e^x. Include the first 4 nonzero terms and the general term.

We know that the power series of exe^x is 1+x+x22!+x33!+...+xnn!+...1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+.... So, since the function is just exe^x being multiplied by x2x^2, we can just multiply the power series of exe^x by x2x^2:

x2ex=x2[1+x+x22!+x33!+...+xnn!+...]x^2e^x=x^2[1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+...]
=x2+x3+x42!+x53!+...+xn+2n!+...=x^2+x^3+\frac{x^4}{2!}+\frac{x^5}{3!}+...+\frac{x^{n+2}}{n!}+...

Great job! Lets look at one more example 🤗

✏️ Power Series Example 2

If h(x)h(x) is the power series centered at x=0x=0 of cos(x)\cos(x), what is h(x)h’(x)? Include the the first 3 nonzero terms and the general term.

We’re given that h(x)=1x22!+x44!+...+(1)nx2n(2n)!+...h(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...+\frac{(-1)^nx^{2n}}{(2n)!}+...To find h(x)h’(x), simply have to take the derivative of the series:

h(x)=x+x33!x55!+...+(1)nx2n1(2n1)!+...h'(x)=-x+\frac{x^3}{3!}-\frac{x^5}{5!}+...+\frac{(-1)^nx^{2n-1}}{(2n-1)!}+...

A cool thing to recognize is that this series is equivalent to the power series of sin(x)\sin(x), but negative; this proves that the derivative of cos(x)\cos(x) is sin(x)-\sin(x).


🔷Practice FRQ 2022 #6

The following free-response question (FRQ) is Question 6 from the 2022 AP Calculus BC examination administered by College Board. All credit to College Board.

The function f is defined by the power series f(x)=xx33+x55x77+...+(1)nx2n+12n+1+...f(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+...+\frac{(-1)^nx^{2n+1}}{2n+1}+... for all real numbers x for which the series converges.

c) Write the first four nonzero terms and the general term for an infinite series that represents f′(x).

For this problem, all you have to do is take the derivative, similar to our example 2:

f(x)=1x2+x4x6+...+(1)nx2n+...f'(x)=1-x^2+x^4-x^6+...+(-1)^nx^{2n}+...

Awesome work! 🎉


👏 Congrats! You’re done.

Congratulations, you’re done with this unit, and as a result, you’ve also reached the end of AP Calculus BC! Now, you have all the tools you need to ace that AP test this May!

As you start to review, be sure to check out our Study Tools unit for more resources and exam information.

Key Terms to Review (20)

Complex Analysis

: Complex analysis is a branch of mathematics that focuses on studying functions involving complex numbers. It explores concepts like holomorphic functions, contour integrals, and residues.

Complex Numbers

: Complex numbers consist of both real and imaginary parts represented by a+bi form (where "a" is the real part and "bi" is the imaginary part). They extend our number system beyond just real numbers.

Differential Equations

: Differential equations are mathematical equations that involve derivatives. They describe how a function changes over time or in relation to other variables.

Epsilon-Delta Definition of a Limit

: The epsilon-delta definition is a rigorous way to define limits in calculus. It states that for every positive value of epsilon (ε), there exists a positive value of delta (δ) such that if the distance between x and c is less than delta, then the distance between f(x) and L is less than epsilon.

Fields

: Fields are mathematical structures that satisfy certain properties, allowing for addition, subtraction, multiplication, and division. They are sets with two operations (addition and multiplication) that follow specific rules.

Fundamental Theorem of Calculus

: The Fundamental Theorem of Calculus states that if f(x) is continuous on an interval [a, b] and F(x) is any antiderivative of f(x), then ∫[a,b] f(x) dx = F(b) - F(a). In simpler terms, it relates differentiation and integration by showing that finding the area under a curve can be done by evaluating its antiderivative at two points.

Geometry

: Geometry is the branch of mathematics that studies shapes, sizes, positions, angles, and dimensions of objects in space. It explores properties such as symmetry, congruence, similarity, and transformations.

Hyperbolic Functions

: Hyperbolic functions are a set of mathematical functions that are analogs to the trigonometric functions. They are defined in terms of exponential functions and can be used to model various physical phenomena.

Linear Algebra

: Linear algebra is a branch of mathematics that deals with vector spaces and linear transformations. It involves studying systems of linear equations, matrices, and their properties.

Maclaurin Polynomial

: A Maclaurin polynomial is a type of Taylor polynomial that is centered at the point x = 0. It is used to approximate a function by adding up terms of different powers of x.

Maclaurin Series

: A Maclaurin series is a special case of Taylor series expansion, where the expansion is centered around x = 0 (or when h = 0).

Multivariable Calculus

: Multivariable calculus is a branch of calculus that deals with functions of multiple variables, where the variables can change simultaneously. It involves studying concepts such as partial derivatives, multiple integrals, and vector fields.

Number Theory

: Number theory is the branch of mathematics that deals with properties and relationships of numbers, particularly integers. It focuses on studying patterns, divisibility, prime numbers, and other number-related concepts.

Power Series

: A power series is an infinite series that represents a function as an infinite polynomial expression.

Reaction Rates

: Reaction rates refer to the speed at which a chemical reaction takes place. It measures how quickly reactants are consumed or products are formed.

Real Analysis

: Real analysis is a branch of mathematics that deals with the study of real numbers and their properties, including limits, continuity, differentiation, and integration.

Taylor Series

: A Taylor series is an expansion of a function into an infinite sum of terms, where each term represents the contribution from different derivatives of the function at a specific point.

Topology

: Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations, such as stretching or bending. It focuses on the concept of "closeness" and the relationships between points, without considering specific distances or measurements.

Trigonometric Substitutions

: Trigonometric substitutions are techniques used to simplify integrals involving radical expressions or quadratic equations. By substituting trigonometric functions for variables, these integrals can be transformed into simpler forms that are easier to solve.

Vector Calculus

: Vector calculus is a branch of mathematics that focuses on studying vector fields and their properties. It involves operations such as differentiation and integration applied to vectors and vector-valued functions.


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.