B Some useful series

For convenience and reference, some results are given here involving series that occur frequently in probability and statistical theory.

B.1 Finite series

Sum of natural numbers:
\[\begin{equation} 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2}. \tag{B.1} \end{equation}\]
Sum of squares of natural numbers:
\[\begin{equation} 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n}{6}(n + 1)(2n + 1). \tag{B.2} \end{equation}\]
Geometric series:
\[\begin{equation} a + ar + ar^2 + \ldots + ar^{n - 1} = \frac{a(1 - r^n)}{1 - r} \tag{B.3}. \end{equation}\] See the next section for the case where $n\to\infty$.
Binomial expansion:
\[\begin{equation} (a + b)^n = b^n + \binom{n}{1}ab^{n - 1} + \ldots + \binom{n}{r}a^r b^{n - r} + \ldots a^n. \tag{B.4} \end{equation}\]

Geometric series:
\[\begin{equation} \sum_{n = 0}^\infty a r^n = a + ar + ar^2 + \ldots \rightarrow \frac{a}{1 - r} \text{ as $n \rightarrow \infty$} \quad\text{for $|r|<1$.} \tag{B.5} \end{equation}\]
Geometric-like series:
\[\begin{equation} \sum_{n = 0}^\infty a n r^n = ar + 2ar^2 + 3ar^3 + \ldots \rightarrow \frac{ar}{(1 - r)^2} \quad\text{for $|r|<1$.} \tag{B.6} \end{equation}\] \[\begin{equation} \sum_{n=0}^\infty a n^2 r^n = ar + 4ar^2 + 9ar^3 \ldots \rightarrow \frac{ar(r + 1)}{(1 - r)^3} \quad\text{for $|r|<1$.} \tag{B.7} \end{equation}\]
Exponential function:
\[\begin{align} e^z &= \sum_{i = 1}^n \frac{z^i}{i!} = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \ldots \quad\text{for all values of $z$}. \tag{B.8}\\ e^{-z} &= \sum_{i = 1}^n \frac{(-z)^i}{i!} = 1 - z + \frac{z^2}{2!} - \frac{z^3}{3!} + \ldots \quad\text{for all values of $z$}. \tag{B.9} \\ e &= \lim_{z\to\infty} \left(1 + \frac{1}{z}\right)^z \tag{B.10}\\ 1/e &= \lim_{z\to\infty} \left(1 - \frac{1}{z}\right)^z \tag{B.11} \end{align}\]
Logarithmic series:
\[\begin{align} \log(1 + z) &= z - \frac{z^2}{2} + \frac{z^3}{3} - \ldots \quad\text{for $|z|<1$}. \tag{B.12}\\ \log(1 - z) &= -z - \frac{z^2}{2} - \frac{z^3}{3} - \ldots \quad\text{for $|z|<1$}. \tag{B.13} \end{align}\]
Others:
\[\begin{align} (1 - z)^{-1} &= 1 + z + z^2 + z^3 + \ldots \quad\text{for $|z|<1$}. \tag{B.14}\\ (1 - z)^{-r} &= 1 + rz + \frac{r(r + 1)z^2}{2!} + \frac{r(r + 1)(r + 2)z^3}{3!} + \ldots \quad\text{for $|z|<1$}. \tag{B.15} \end{align}\]

Note that the expressions on the RHS above do not converge to the LHS for all values of $z$.