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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 Some careful addition shows it is \(\frac{46181}{176400}\text{.}\); 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 Or possibly gobbledygook. For a discussion of statements without meaning and why one should avoid them we recommend the book “Bendable learnings: the wisdom of modern management” by Don Watson. .