C.1 Summations and Series

Summations and Series are an important part of discrete probability theory. We provide a brief review of some of the series used in STAT 414. While it is important to recall these special series, you should also take the time to practice. For more in depth review, there are links to Khan Academy.

Summations

First, it is important to review the notation. The symbol, \(\sum\), is a summation. Suppose we have the sequence, \(a_1, a_2, \cdots, a_n\), denoted \(\{a_n\}\), and we want to sum all their values. This can be written as

\[\sum_{i=1}^n a_i\]

Here are some special sums:

  1. \(\sum_{i=1}^n i=1+2+\cdots+n=\frac{n(n+1)}{2}\)
  2. \(\sum_{i=1}^n i^2=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}\)
  3. The Binomial Theorem:

It is possible to expand any power of \(x+y\) to the sum

\[(x+y)^n=\sum_{i=0}^n {n \choose i} x^{n-i}y^i\]

where

\[{n\choose i}=\frac{n(n-1)(n-2)\cdots(n-i-1)}{i!}=\frac{n!}{(n-i)!i!}\]

Examples using the Binomial Theorem Video, (Khan Academy).


Series

When n is a finite number, the value of the sum can be easily determined. How do we find the sum when the sequence is infinite? For example, suppose we have an infinite sequence, \(a_1, a_2, \cdots\). The infinite series is denoted:

\[S=\sum_{i=1}^\infty a_i\]

For infinite series, we consider the partial sums. Some partial sums are

\[\begin{align*}
& S_1=\sum_{i=1}^1 a_i=a_1 \\
& S_2=\sum_{i=1}^2 a_i=a_1+a_2 \\
& S_3=\sum_{i=1}^3 a_i=a_1+a_2+a_3\\
& \vdots\\
& S_n=\sum_{i=1}^n a_i=a_1+a_2+\cdots+a_n
\end{align*}\]

An infinite series converges and has sum S if the sequence of partial sums, \(\{S_n\}\) converges to S. Thus, if

\[S=\lim_{n\rightarrow \infty} \{S_n\}\]

then the series converges to S. If \(\{S_n\}\) diverges, then the series diverges.

Review Convergence and Divergence of Series Video, (Khan Academy).


These are some of the special series used in STAT 414. It would be helpful to review more than what is listed below.

Geometric series

A geometric series has the form

\[S=\sum_{k=1}^\infty a r^{k-1}=a+ar+ar^2+ar^3+\cdots\]

where \(a\neq 0\). A geometric series converges to \(\frac{a}{1-r}\) if \(|r|<1\), but diverges if \(|r|\ge1\).

More examples and Explanation of the Geometric Series Video, (Khan Academy).

A special case of the geometric series

\[\frac{1}{1-x}=1+x+x^2+x^3+\cdots\]

for $-1<x<1$.

The Taylor (or Maclaurin) series of \(e^x\):

The series:

\[\sum_{i=0}^\infty \frac{x^i}{i!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots\]

for \(-1\le x\le 1\) converges to \(e^x\).

Review for the Taylor (or Maclaurin) Series Video, (Khan Academy).

Example C.1

Does the series \(\frac{1}{3}-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\cdots\) converge? If so, to what limit?

\[S=\frac{1}{3}-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\cdots=\sum_{x=0}^{\infty} \frac{1}{3(-2)^x}\]

This is a geometric series with \(a=\frac{1}{3}\) and \(r=-\frac{1}{2}\). Therefore, it converges to

\[\frac{a}{1-r}=\frac{\frac{1}{3}}{1+\frac{1}{2}}=\frac{2}{9}\]