A Some useful series
For convenience and reference, some results are given here involving series that occur frequently in probability and statistical theory.
A.1 Finite series
Sum of natural numbers:
1+2+3+…+n=n(n+1)2.Sum of squares of natural numbers:
12+22+32+…+n2=n6(n+1)(2n+1).Geometric series:
a+ar+ar2+…+arn−1=a(1−rn)1−r. See the next section for the case where n→∞.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{A.4} \end{equation}
A.2 Infinite series and limits
Geometric series:
\begin{equation} \sum_{n = 0}^\infty a r^n = a + ar + ar^2 + \ldots \rightarrow \frac{a}{1 - r} \text{ as $n \rightarrow \infty$} \tag{A.5} \end{equation} where the infinite series only converges if |r| < 1.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} \tag{A.6} \end{equation} where the infinite series only converges if |r| < 1.
\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} \tag{A.7} \end{equation} where the infinite series only converges if |r| < 1.Exponential function:
\begin{align} e^z &= 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \ldots \tag{A.8}\\ e^{-z} &= 1 - z + \frac{z^2}{2!} - \frac{z^3}{3!} + \ldots \tag{A.9} \\ e &= \lim_{n\to\infty} \left(1 + \frac{1}{z}\right)^z \tag{A.10}\\ 1/e &= \lim_{n\to\infty} \left(1 - \frac{1}{z}\right)^z \tag{A.11} \end{align}Logarithmic series:
\begin{align} \log(1 + z) &= z - \frac{z^2}{2} + \frac{z^3}{3} - \ldots \tag{A.12}\\ \log(1 - z) &= -z - \frac{z^2}{2} - \frac{z^3}{3} - \ldots \tag{A.13} \end{align}Others:
\begin{align} (1 - z)^{-1} &= 1 + z + z^2 + z^3 + \ldots \tag{A.14}\\ (1 - z)^{-r} &= 1 + rz + \frac{r(r + 1)z^2}{2!} + \frac{r(r + 1)(r + 2)z^3}{3!} + \ldots \tag{A.15} \end{align}
Note that the expressions on the RHS above do not converge to the LHS for all values of z.