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Questions and Answers
What is the expansion for $(1+x)ⁿ$ according to the given series?
What is the series representation for $e^x$?
Which expression represents the power series of $ln(1+x)$ over the specified range?
What is the general form of the expression $a^x$ in a series expansion?
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What is the identity for $e^{ix}$ derived from Euler's formula?
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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).
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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.
