## Taylor Series

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 $$a = 0$$.

### Geometric Series

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:

$\sum_{k = m}^{\infty} x^k = \frac{x^{m}}{1 - x}$

### Exponential Function

$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$

Or, more compactly:

$\sum_{k = 0}^{\infty} \frac{x^{k}}{k!} = e^{x}$

## Binomial Expansion

$(a + b)^{n} = \sum_{k = 0}^{n} \binom{n}{k}a^{n - k}b^{k}$

## Sum of Power of Intergers

$\sum_{k = 1}^{n} k = \frac{n(n+1)}{2}$

$\sum_{k = 1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$