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Section3.5Summation Formulas

Subsection3.5.1Overview

When an increment sequence is given explicitly, we often seek to find an explicit formula for the accumulation sequence. This section introduces basic properties of summation as well as some basic formulas for a few elementary increment sequences. Combining the basic formulas with the properties of summation, we can compute explicit formulas for accumulations when the increment sequence is a linear combination of these elementary sequences. To prove that these formulas are correct, we also introduce a theorem that shows that a sequence is uniquely defined by its initial value and increments.

Subsection3.5.2Properties of Summation

Because a summation involves only addition, the basic properties of addition translate to corresponding properties for summation.

Suppose we have a sequence \(x\) and a constant \(\alpha\). We can create a new sequence \(\alpha x\), called a constant multiple, by multiplying every term of \(x\) by the same constant \(\alpha\). Using the constant multiple as an increment sequence, every term will have a common factor of \(\alpha\). This leads to a property of summation called the constant multiple rule.

The next property considers a sequence that is itself formed by adding two sequences together. Suppose we have two sequences \(u\) and \(w\) and we form a new sequence \(u+w\) with values that are the sum of the corresponding values of \(u\) and \(w\). Because addition is both commutative and associative, any sum of a finite number of terms can be regrouped in any convenient way. A summation of terms \(u+w\) can therefore be grouped in a way that we add only the terms from \(u\) and then add only the terms from \(v\) and then add the results. This leads to a property of summation called the sum rule.

Using the rules together creates a new rule called linearity involving two sequences \(x\) and \(y\). The idea for this rule is that an individual term in the increment sequence is the sum of a constant multiple of each, \(\alpha x + \beta y\). Such a sum is called a linear combination of \(x\) and \(y\) with coefficients \(\alpha\) and \(\beta\). This name results from the general equation of a line being of the form \(ax + by = c\).

Using \(\alpha = 1\) and \(\beta = -1\), the linear combination becomes a difference, \(\alpha x + \beta y = x - y\). So the difference rule is a special case of linearity.

There are no corresponding rules for multiplication or division. This is really no different than emphasizing the importance of multiplying all terms using the distributive property, such as occurs with the FOIL method for multiplying binomials. For example, \(\displaystyle \sum_{k=1}^{3} [k] = 1+2+3 = 6\). The product of the sum gives one result: \begin{equation*} \sum_{k=1}^{3}[k] \cdot \sum_{k=1}^{3}[k] = (1+2+3) \cdot (1+2+3) = 6 \cdot 6 = 36. \end{equation*} But the sum of the products gives a different result: \begin{equation*} \sum_{k=1}^{3}[k \cdot k] = (1^2+2^2+3^2) = 1+4+9 = 14. \end{equation*} In general, \begin{equation*} \sum_{k=m}^{n} [x_k \cdot y_k] \ne \sum_{k=m}^{n} x_k \cdot \sum_{k=m}^{n} y_k. \end{equation*}

The other important properties of summation focus on how to split a summation into two pieces. Suppose a summation has lower index limit \(m\) and upper index limit \(n\). The total summation can be split into two parts, where the first part is a summation involving the terms having index values from \(m\) to \(k\) and the second part is a summation involving the remaining terms with index values \(k+1\) through \(n\). (Of course, since we chose to use \(k\) as that intermediate integer, our dummy variable for the summation needs to be different.)

A consequence of the splitting property of summation is that a summation over a particular range of index values can be written as a difference of a summation over a larger range of index values minus summations representing the terms that were not in the original range of summation.

In order to establish formulas for summations, we will need a way to demonstrate that two sequences are really the same. We conclude this subsection by introducing a theorem that is not so much a property of summation, but a method to infer that sequences from two different representations are actually the same.

Proof

Theorem 3.4.11 stated that every sequence is an accumulation of its backward differences. This is an immediate corollary of Theorem 3.5.7. But we will be able to use this theorem in many other circumstances when we wish to show that two representations of a sequence give the same sequence.

Subsection3.5.3Elementary Summation Formulas

There are some elementary increment sequences for which we can find an explicit formula for the accumulation sequence. We will state the results and prove them using the uniqueness criteria for sequences.

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Example3.5.13

Find the sum of the integers \(1, 2, \ldots, 100\).

Solution
Example3.5.14

Find the sum of the integers \(100, 101, \ldots, 200\).

Solution

Subsection3.5.4Summations of Combinations

With the accumulation of elementary sequences established and theorems regarding combinations of sequences proved, we are ready to apply our summation formulas for more complex summations that involve linear combinations of elementary sequences. Recall that a linear combination of sequences is the sum of constant multiples of those other sequences. Only sums behave well with summation. If a summation involves a product or quotient (or other complex function) of elementary sequences, then the strategy described here does not apply.

The basic idea introduced here is to apply the linearity property of summations stated in Theorem 3.5.3. The sums will split into separate summations and the constant multiples factor out of the individual sums. We will then apply our known summation formulas for the remaining summations to find our desired explicit formula.

Example3.5.15

Find \(\displaystyle \sum_{k=1}^{20} (500+60k-2k^2)\).

Solution

The same strategy still applies if the constant multiple coefficients are written using parameters or even using variables other than the dummy index variable of summation. In particular, when the upper limit of the summation is a variable, the formula for the sequence might also involve that variable as well as the index variable. Because this will be encountered frequently, an example is provided below.

Example3.5.16

Find a formula for \(\displaystyle \sum_{k=1}^{n} \left( \frac{3k}{n^2} - \frac{k^2}{n^3} \right)\) that involves only \(n\).

Solution