Subsection 3.4.4 Whirlwind Tour of Summation Notation
¶In the remainder of this section we will frequently need to write sums involving a large number of terms. Writing out the summands explicitly can become quite impractical — for example, say we need the sum of the first 11 squares:
\begin{gather*}
1 + 2^2 + 3^2 + 4^2+ 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2
\end{gather*}
This becomes tedious. Where the pattern is clear, we will often skip the middle few terms and instead write
\begin{gather*}
1 + 2^2 + \cdots + 11^2.
\end{gather*}
A far more precise way to write this is using \(\Sigma\) (capital-sigma) notation. For example, we can write the above sum as
\begin{gather*}
\sum_{k=1}^{11} k^2
\end{gather*}
This is read as
The sum from \(k\) equals 1 to 11 of \(k^2\text{.}\)
More generally
Definition 3.4.8
Let \(m\leq n\) be integers and let \(f(x)\) be a function defined on the integers. Then we write
\begin{gather*}
\sum_{k=m}^n f(k)
\end{gather*}
to mean the sum of \(f(k)\) for \(k\) from \(m\) to \(n\text{:}\)
\begin{gather*}
f(m) + f(m+1) + f(m+2) + \cdots + f(n-1) + f(n).
\end{gather*}
Similarly we write
\begin{gather*}
\sum_{i=m}^n a_i
\end{gather*}
to mean
\begin{gather*}
a_m+a_{m+1}+a_{m+2}+\cdots+a_{n-1}+a_n
\end{gather*}
for some set of coefficients \(\{ a_m, \ldots, a_n \}\text{.}\)
Consider the example
\begin{gather*}
\sum_{k=3}^7 \frac{1}{k^2}=\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+
\frac{1}{6^2}+\frac{1}{7^2}
\end{gather*}
It is important to note that the right hand side of this expression evaluates to a number 6 ; it does not contain “\(k\)”. The summation index \(k\) is just a “dummy” variable and it does not have to be called \(k\text{.}\) For example
\begin{gather*}
\sum_{k=3}^7 \frac{1}{k^2}
=\sum_{i=3}^7 \frac{1}{i^2}
=\sum_{j=3}^7 \frac{1}{j^2}
=\sum_{\ell=3}^7 \frac{1}{\ell^2}
\end{gather*}
Also the summation index has no meaning outside the sum. For example
\begin{gather*}
k\sum_{k=3}^7 \frac{1}{k^2}
\end{gather*}
has no mathematical meaning; It is gibberish 7 .