Binomial and Exponential Series

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).