Glossary

Binomial Series

Criticality: 2

A specific Maclaurin series for functions of the form $(1+x)^a$, where 'a' can be any real number.

Example:

The binomial series can be used to expand $\sqrt{1+x}$ as $1 + \frac{1}{2}x - \frac{1}{8}x^2 + ...$ .

Geometric Series

Criticality: 3

A series where each term after the first is found by multiplying the previous one by a fixed, non-zero common ratio. In calculus, it often refers to the power series $\sum x^n = \frac{1}{1-x}$.

Example:

The Maclaurin series for $\frac{1}{1-x}$ is a geometric series $1+x+x^2+x^3+...$ , which converges for $|x|<1$ .

Maclaurin Series

Criticality: 3

A special case of a Taylor series where the series is centered specifically at $x=0$.

Example:

The Maclaurin series for $\cos(x)$ is $1-\frac{x^2}{2!}+\frac{x^4}{4!}-...$ , which is a Taylor series centered at the origin.

Pattern Recognition

Criticality: 2

The skill of identifying a general formula for the $n$-th derivative of a function, which is crucial for constructing the general form of a Taylor or Maclaurin series.

Example:

When finding the Taylor series for $f(x) = e^{2x}$ , pattern recognition helps us see that $f^{(n)}(x) = 2^n e^{2x}$ .

Power Series

Criticality: 3

An infinite series of the form $\sum_{n=0}^\infty c_n (x-a)^n$, which represents a function as a sum of terms involving powers of $(x-a)$.

Example:

The geometric series $1+x+x^2+x^3+...$ is a power series representation for $\frac{1}{1-x}$ centered at $x=0$ .

Taylor Approximations Theorem

Criticality: 2

A theorem that allows functions to be approximated as polynomials, forming the foundational idea behind Taylor polynomials and series.

Example:

Using the Taylor approximations theorem, we can approximate $\sin(x)$ near $x=0$ with a polynomial like $x - \frac{x^3}{6}$ .

Taylor Polynomial

Criticality: 3

A finite partial sum of a Taylor series, used to approximate a function up to a certain degree.

Example:

The third-degree Taylor polynomial for $e^x$ centered at $x=0$ is $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}$ , providing a good approximation for $e^x$ near zero.

Taylor Series

Criticality: 3

A representation of a function as an infinite sum of terms, calculated from the values of the function's derivatives at a single point, known as the center.

Example:

The Taylor series for $e^x$ centered at $x=0$ is $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$ , which allows us to approximate $e^x$ for any $x$ .

Glossary

Binomial Series

Criticality: 2

A specific Maclaurin series for functions of the form $(1+x)^a$, where 'a' can be any real number.

Example:

The binomial series can be used to expand $\sqrt{1+x}$ as $1 + \frac{1}{2}x - \frac{1}{8}x^2 + ...$ .

Geometric Series

Criticality: 3

A series where each term after the first is found by multiplying the previous one by a fixed, non-zero common ratio. In calculus, it often refers to the power series $\sum x^n = \frac{1}{1-x}$.

Example:

The Maclaurin series for $\frac{1}{1-x}$ is a geometric series $1+x+x^2+x^3+...$ , which converges for $|x|<1$ .

Maclaurin Series

Criticality: 3

A special case of a Taylor series where the series is centered specifically at $x=0$.

Example:

The Maclaurin series for $\cos(x)$ is $1-\frac{x^2}{2!}+\frac{x^4}{4!}-...$ , which is a Taylor series centered at the origin.

Pattern Recognition

Criticality: 2

The skill of identifying a general formula for the $n$-th derivative of a function, which is crucial for constructing the general form of a Taylor or Maclaurin series.

Example:

When finding the Taylor series for $f(x) = e^{2x}$ , pattern recognition helps us see that $f^{(n)}(x) = 2^n e^{2x}$ .

Power Series

Criticality: 3

An infinite series of the form $\sum_{n=0}^\infty c_n (x-a)^n$, which represents a function as a sum of terms involving powers of $(x-a)$.

Example:

The geometric series $1+x+x^2+x^3+...$ is a power series representation for $\frac{1}{1-x}$ centered at $x=0$ .

Taylor Approximations Theorem

Criticality: 2

A theorem that allows functions to be approximated as polynomials, forming the foundational idea behind Taylor polynomials and series.

Example:

Using the Taylor approximations theorem, we can approximate $\sin(x)$ near $x=0$ with a polynomial like $x - \frac{x^3}{6}$ .

Taylor Polynomial

Criticality: 3

A finite partial sum of a Taylor series, used to approximate a function up to a certain degree.

Example:

The third-degree Taylor polynomial for $e^x$ centered at $x=0$ is $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}$ , providing a good approximation for $e^x$ near zero.

Taylor Series

Criticality: 3

A representation of a function as an infinite sum of terms, calculated from the values of the function's derivatives at a single point, known as the center.

Example:

The Taylor series for $e^x$ centered at $x=0$ is $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$ , which allows us to approximate $e^x$ for any $x$ .