Some Important Infinite Series

We are about to look at a very important method to solving certain counting problems with the use of special functions know as Generating Functions. We will first need to review some important infinite series first. More explanation on these series can be found on the Calculus page.

• Convergent Geometric Series: For $-1 < x < 1$:

(1)

\begin{align} \quad \sum_{i=0}^{\infty} x^i = 1 + x + x^2 + ... + x^i + ... = \frac{1}{1 - x} \end{align}

• Derivative of Convergent Geometric Series: For $-1 < x < 1$:

(2)

\begin{align} \quad \sum_{i=0}^{\infty} ix^{i-1} = 1 + 2x + 3x^2 + ... + ix^{i-1} + ... = \frac{1}{(1 - x)^2} \end{align}

• Antiderivative of Convergent Geometric Series: For $-1 \leq x < 1$:

(3)

\begin{align} \quad \sum_{i=0}^{\infty} \frac{x^{i+1}}{i+1} = x + \frac{x^2}{2} + \frac{x^3}{3} + ... + \frac{x^{i+1}}{i+1} + ... = -\ln (1 - x) \end{align}

• Exponential Series: For all $x \in \mathbb{R}$:

(4)

\begin{align} \quad \sum_{i=0}^{\infty} \frac{x^i}{i!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... + \frac{x^i}{i!} + ... = e^x \end{align}

(5)

\begin{align} \quad \sum_{i=0}^{\infty} (-1)^i \frac{x^i}{i!} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + ... + (-1)^i \frac{x^i}{i!} + ... = e^{-x} \end{align}

(6)

\begin{align} \quad \sum_{i=0}^{\infty} \frac{x^{2i}}{(2i)!} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + ... + \frac{x^{2i}}{(2i)!} + ... = \frac{e^x + e^{-x}}{2} = \cosh x \end{align}