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Glossary

B

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 1+x\sqrt{1+x} as 1+12x−18x2+...1 + \frac{1}{2}x - \frac{1}{8}x^2 + ....

G

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 11−x\frac{1}{1-x} is a geometric series 1+x+x2+x3+...1+x+x^2+x^3+..., which converges for ∣x∣<1|x|<1.

M

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)\cos(x) is 1−x22!+x44!−...1-\frac{x^2}{2!}+\frac{x^4}{4!}-..., which is a Taylor series centered at the origin.

P

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)=e2xf(x) = e^{2x}, pattern recognition helps us see that f(n)(x)=2ne2xf^{(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+x2+x3+...1+x+x^2+x^3+... is a power series representation for 11−x\frac{1}{1-x} centered at x=0x=0.

T

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)\sin(x) near x=0x=0 with a polynomial like x−x36x - \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 exe^x centered at x=0x=0 is 1+x+x22!+x33!1+x+\frac{x^2}{2!}+\frac{x^3}{3!}, providing a good approximation for exe^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 exe^x centered at x=0x=0 is 1+x+x22!+x33!+...1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+..., which allows us to approximate exe^x for any xx.