File Name: summation and product notation .zip
In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a definition, which, while intimidating, is meant to make our lives easier. The lower and upper limits of the summation tells us which term to start with and which term to end with, respectively. For instance,. One place you may encounter summation notation is in mathematical definitions. For example, summation notation allows us to define polynomials as functions of the form.
In this section we need to do a brief review of summation notation or sigma notation. For large lists this can be a fairly cumbersome notation so we introduce summation notation to denote these kinds of sums. The case above is denoted as follows. In other words,. Here are a couple of nice formulas that we will find useful in a couple of sections. You can, of course, derive other formulas from these for different starting points if you need to.
This formula shows how a finite sum can be split into two finite sums. This formula shows that a constant factor in a summand can be taken out of the sum. This formula represents the concept that the sum of logs is equal to the log of the product , which is correct under the given restriction. This formula is called the Dirichlet formula for a Fourier series. In this formula, the sum is divided into the sums of the even and odd terms. In this formula, the sum of is divided into three sums with the terms , , and.
The Sigma symbol, , is a capital letter in the Greek alphabet. The Sigma symbol can be used all by itself to represent a generic sum… the general idea of a sum, of an unspecified number of unspecified terms:. But this is not something that can be evaluated to produce a specific answer, as we have not been told how many terms to include in the sum, nor have we been told how to determine the value of each term. All that matters in this case is the difference between the starting and ending term numbers… that will determine how many twos we are being asked to add, one two for each term number. Sigma notation, or as it is also called, summation notation is not usually worth the extra ink to describe simple sums such as the one above… multiplication could do that more simply.
Given a sequence a 1 , a 2 , The value of a finite series is always well defined, and its terms can be added in any order. If the limit does not exist, the series diverges ; otherwise, it converges. The terms of a convergent series cannot always be added in any order. We can, however, rearrange the terms of an absolutely convergent series , that is, a series for which the series also converges.
For reference, this section introduces the terminology used in some texts to describe the minterms and maxterms assigned to a Karnaugh map. Otherwise, there is no new material here. The following example is revisited to illustrate our point. Instead of a Boolean equation description of unsimplified logic, we list the minterms. The numbers indicate cell location, or address, within a Karnaugh map as shown below right. This is certainly a compact means of describing a list of minterms or cells in a K-map. The Sum-Of-Products solution is not affected by the new terminology.
Worksheet: Sum/Product Notations Solutions Express the following sums in summation notation and evaluate using appropriate summation formulas. (a) 3 + 7.
At times when we add, there is a pattern by which we can express the addends. For instance, in the sum. Likewise, in the sum. See whether you can detect and describe the addend patterns in the following sums.
notations are abbreviations for repeated sums or repeated products, and it is Be familiar with the standard summation formulas: The following are formulas that.