Binomial and Exponential Series

Study Notes

Binomial Expansion

The binomial expansion of ((1+x)^{n}) illustrates how polynomial expressions can be expressed as a series, applicable for (x \in \mathbb{R}).
The series includes terms like (1 + n x + \frac{n(n-1)x^2}{2!} + \frac{n(n-1)(n-2)x^3}{3!} + \ldots).

Exponential Series

The exponential function (e^x) can be expanded as the infinite series (1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots), valid for all (x \in \mathbb{R}).

General Exponential Function

For the function (a^x), the series expansion is (1 + \frac{x \ln{a}}{1!} + \frac{(x \ln{a})^2}{2!} + \ldots), applicable for all (x).

Natural Logarithm Expansion

The logarithmic function (\ln(1+x)) can be expressed as a power series: (x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots), which converges for (-1 \leq x \leq 1).

Euler's Formula

Euler's formula states that (e^{ix} = \cos x + i \sin x) establishes a deep relationship between exponential functions and trigonometric functions.
The equation is valid for (x \to i\infty).

Taylor Series for Complex Exponential

The expansion for (e^{ix}) using Taylor series is (1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \ldots) as (x \to \infty).
The terms separate into real and imaginary parts yielding (\cos x + i \sin x), with (\cos x) involving even derivatives and (\sin x) involving odd derivatives.

Series Representation of Trigonometric Functions

The series for (\cos x) includes terms (1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \ldots).
The series for (\sin x) is composed of odd-powered terms: (x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots).

Description

This quiz explores the concepts of binomial expansion and various series expansions, including Euler's formula and natural logarithm series. Test your understanding of how polynomial expressions and exponential functions can be represented as series, applicable for real numbers.