For any function \(f(x)\), the Taylor series of \(f(x)\) at \(a\) is:
\[ f(x) = f(a) + (x - a)f'(a) + \frac{(x - a)^2}{2!}f''(a) + \cdots + \frac{(x - a)^n}{n!}f^{(n)}(a) + \cdots \]
Or, more compactly:
\[ f(x) = \sum_{n = 0}^{\infty} \frac{(x - a)^n}{n!}f^{(n)}(a) \]
In this course, we will often use two Maclaurin series, which are Taylor series with \(a = 0\).
Provided \(|x| < 1\),
\[ \frac{1}{1 - x} = 1 + x + x^2 + x^3 + \cdots. \]
Or, more compactly:
\[ \sum_{k = 0}^{\infty} x^k = \frac{1}{1 - x} \]
We could also start from an arbitrary index:
\[ \sum_{k = m}^{\infty} x^k = \frac{x^{m}}{1 - x} \]
\[ e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots \]
Or, more compactly:
\[ \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} = e^{x} \]
\[ (a + b)^{n} = \sum_{k = 0}^{n} \binom{n}{k}a^{n - k}b^{k} \]
\[ \sum_{k = 1}^{n} k = \frac{n(n+1)}{2} \]
\[ \sum_{k = 1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \]