Maclaurin Series

The Maclaurin series is a special case of the Taylor series expanded about \( x = 0 \). It provides a way to represent functions as infinite sums of powers of \( x \).

The Maclaurin series of a function \( f(x) \) is given by:

\(\boxed{ f(x) = f(0) + f'(0) x + \frac{f”(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 + \cdots = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n }\)

where \( f^{(n)}(0) \) is the \( n^{th} \) derivative of \( f(x) \) evaluated at \( x=0 \).

Below are some common Maclaurin series expansions:

1. Exponential function \( e^x \):

\( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^\infty \frac{x^n}{n!} \)

2. Sine function \( \sin x \):

\( \sin x = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} \)

3. Cosine function \( \cos x \):

\( \cos x = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!} \)

4. Natural logarithm \( \ln(1+x) \), valid for \( -1 < x \leq 1 \):

\( \ln(1+x) = x – \frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \cdots = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n} \)

5. General binomial expansion \( (1+x)^p \), for any rational \( p \in \mathbb{Q} \), valid for \( |x| < 1 \):

$(1+x)^p = 1 + p x + \frac{p(p-1)}{2!} x^2 + \frac{p(p-1)(p-2)}{3!} x^3 + \cdots = \sum_{n=0}^\infty \binom{p}{n} x^n $

where the generalized binomial coefficient is \( \displaystyle \binom{p}{n} = \frac{p(p-1)(p-2) \cdots (p-n+1)}{n!} \), with \( \binom{p}{0} = 1 \).