Summation and product notation pdf

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summation and product notation pdf

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At times when we add, there is a pattern by which we can express the addends. For instance, in the sum.

Sigma and Pi Notation (Summation and Product Notation)

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. Summation notation provides for us a compact way to represent the addends in sums such as these. For instance, here is the summation notation to represent the sum of the first 10 positive integers, the first sum described on this page.

The annotated symbolism shown below identifies important elements used in summation notation also called sigma notation.

To expand this summation notation, that is, to determine the set of addends that we are to sum, we replace any occurance of the dummy variable in the addend representation with the lower limit of the index variable. We evaluate the resulting expression. This is our first addend. We repeat this process with the next value of the index variable, using that specific value for the index variable in the addend representation and simplifying as desired or necessary.

The replace and simplify process continues until the last index value to be used is the upper limit of summation.

Each addend in the sum will be the square of an index value. The index values begin with 3 and increase by 1 until reaching 7. Thus, we have the index values 3, 4, 5, 6, and 7, and the squares of those are 9, 16, 25, 36, and In some cases we may not identify the upper limit of summation with a specific value, instead usingf a variable.

Here's an example. The lower limit of summation is 0 and the upper limit is n. Each addend in the sum is found by multiplying the index value by 3 and then adding 1 to that.

Because we do not know the specific value for n, we use an elipsis. Here's the expansion of this summation notation. We may also create sums with an infinite number of addends. In this situation, the upper limit of summation is infinity.

There is no last addend, because the upper limit of summation is infinity, indicating we simply continue to create addends following the pattern shown. Here's the expansion for this infinite summation. Once you've learned how to use summation notation to express patterns in sums, product notation has many similar elements that make it straightforward to learn to use.

The only difference is that we use product notation to express patterns in products, that is, when the factors in a product can be represented by some pattern. Instead of the Greek letter sigma, we use the Greek letter pi. Here is product notation to represent the product of the first several squares:. Summation and Product Notation. Session Outlines. Assignments and Problem Sets.

Sum and Product Notation

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. In this formula, the sum of is divided into four sums with the terms , , , and.

Worksheet: Sum/Product Notations Solutions Express the following sums in summation notation and evaluate using appropriate summation formulas. (a) 3 + 7.

Summation and the sigma notation

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.

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.

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.

9.2: Summation Notation

Summation and the sigma notation

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.

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.

Sum and Product Notation: Delimited Forms. The capital Greek letter Σ (sigma) denotes summation. Sums and Products: General Sigma and Pi Notations.

Sum and Product Notation


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