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The Fundamental Theorem of Calculus
Three Different ConceptsThe Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1
The Indefinite Integral and the Net Change
Indefinite Integrals and Anti-derivativesA Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy
Substitution
Substitution for Indefinite IntegralsExamples to Try
Revised Table of Integrals
Substitution for Definite Integrals
Examples
Area Between Curves
Computation Using IntegrationTo Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary
Volumes
Slicing and Dicing SolidsSolids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice
Integration by Parts
Integration by PartsExamples
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig Functions
Antiderivatives of Basic Trigonometric FunctionsProduct of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (only even powers)
Product of Secants and Tangents
Other Cases
Trig Substitutions
How Trig Substitution WorksSummary of trig substitution options
Examples
Completing the Square
Partial Fractions
IntroductionLinear Factors
Irreducible Quadratic Factors
Improper Rational Functions and Long Division
Summary
Strategies of Integration
SubstitutionIntegration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions
Improper Integrals
Type 1 - Improper Integrals with Infinite Intervals of IntegrationType 2 - Improper Integrals with Discontinuous Integrands
Comparison Tests for Convergence
Differential Equations
IntroductionSeparable Equations
Mixing and Dilution
Models of Growth
Exponential Growth and DecayLogistic Growth
Infinite Sequences
Approximate Versus Exact AnswersExamples of Infinite Sequences
Limit Laws for Sequences
Theorems for and Examples of Computing Limits of Sequences
Monotonic Covergence
Infinite Series
IntroductionGeometric Series
Limit Laws for Series
Test for Divergence and Other Theorems
Telescoping Sums and the FTC
Integral Test
Road MapThe Integral Test
Estimates of Value of the Series
Comparison Tests
The Basic Comparison TestThe Limit Comparison Test
Convergence of Series with Negative Terms
Introduction, Alternating Series,and the AS TestAbsolute Convergence
Rearrangements
The Ratio and Root Tests
The Ratio TestThe Root Test
Examples
Strategies for testing Series
Strategy to Test Series and a Review of TestsExamples, Part 1
Examples, Part 2
Power Series
Radius and Interval of ConvergenceFinding the Interval of Convergence
Power Series Centered at $x=a$
Representing Functions as Power Series
Functions as Power SeriesDerivatives and Integrals of Power Series
Applications and Examples
Taylor and Maclaurin Series
The Formula for Taylor SeriesTaylor Series for Common Functions
Adding, Multiplying, and Dividing Power Series
Miscellaneous Useful Facts
Applications of Taylor Polynomials
Taylor PolynomialsWhen Functions Are Equal to Their Taylor Series
When a Function Does Not Equal Its Taylor Series
Other Uses of Taylor Polynomials
Partial Derivatives
Visualizing Functions in 3 DimensionsDefinitions and Examples
An Example from DNA
Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions
Multiple Integrals
BackgroundWhat is a Double Integral?
Volumes as Double Integrals
Iterated Integrals over Rectangles
How To Compute Iterated IntegralsExamples of Iterated Integrals
Cavalieri's Principle
Fubini's Theorem
Summary and an Important Example
Double Integrals over General Regions
Type I and Type II regionsExamples 1-4
Examples 5-7
Order of Integration
Functions as Power Series
A power series $\displaystyle\sum_{n=0}^\infty c_n x^n$ can be
thought of as a function of $x$ whose domain is the interval of
convergence.
Conversely, many functions can be expressed as power series,
and we will be learning various ways to do this. This is
extremely valuable; for example, $e^x$ can be expressed as a power
series, which is really just an infinite polynomial. The
power series is easy to estimate by evaluating out to as many
terms as you wish.
Consider $\displaystyle\sum_{n=0}^\infty r^n$. This is geometric series converges when $|r|<1$ and diverges otherwise. When it converges, its value is $\frac{1}{1-r}$. We take that series, and replace $r$ with $x$, getting a power series. We can express this as a function, as you see below below, as long as $|x|<1$.
$$f(x)=\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1+x+x^2+x^3+x^4+\cdots\qquad\text{ as long as }\lvert x\rvert<1$$
We get a different function by replacing $x$ with $-x$: $$\displaystyle{g(x)=\frac{1}{1+x} = \sum_{n=0}^\infty (-x)^n = 1-x+x^2+\cdots}\quad\text{ as long as } \lvert -x\rvert=\lvert x\rvert<1.$$
The video will look at these and other examples arising from the geometric series.
DO: Work on the following two examples before reading ahead.
Example 1: Find a power series representation of the function $\displaystyle\frac{x}{1+x^2}$, and determine for which $x$ it would be defined.
Example 2: Find a power series representation of the function $\displaystyle\frac{1}{7+2x}$, and determine for which $x$ it would be defined.
Solution 1: Replace $x$ (in our original $f(x)$ before the video) by $-x^2$, and multiply the expression by $x$. $$\displaystyle{\frac{x}{1+x^2} = x\sum_{n=0}^\infty (-x^2)^n =\sum_{n=0}^\infty x(-x^2)^n=\sum_{n=0}^\infty (-1)^n x^{2n+1}= x-x^3+x^5-x^7+\cdots}.$$For convergence, we need $|x^2|<1$, which simplifies to $\lvert x\rvert<1$.
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Solution 2: Divide out a $7$ in the denominator, in order to have the constant equal to 1: $\displaystyle\frac{1}{7+2x}=\frac{1}{7}\frac{1}{1+\left(\frac{2x}7\right)}.$ Now we can see that we replace our original $x$ by $-\frac{2x}{7}$ and multiply the expression by $\frac17$.
$$\displaystyle \frac{7x}{1+2x}=\left(\frac17\right)\sum_{n=0}^\infty\left(-\frac{2x}{7}\right)^n=\left(\frac17\right)\sum_{n=0}^\infty(-1)^n\left(\frac{2x}7 \right)^n=\sum_{n=0}^\infty(-1)^n\frac{(2x)^n}{7^{n+1}}=\frac17-\frac{2}{7^2}x+\frac{2^2}{7^3}x^2+\frac{2^3}{7^4}x^3+\cdots .$$For convergence, we need $\lvert\frac{2x}{7}\rvert<1$. This simplifies to $|x|<\frac72$.