![]() ![]() ![]() Inthis situation, the upper limit of summation is infinity. We may also create sums with an infinite number of addends. Because we do not know the specific value for n, we usean elipsis (.) to signal that the addend pattern continues.Here's the expansion of this summation notation. When we reach the upper limit, the addendis (3)(n)+1. Eachaddend in the sum is found by multiplying the index value by 3 andthen adding 1 to that. The lower limit of summation is 0 and the upper limit is n. In some cases we may not identify the upper limit of summationwith a specific value, instead usingf a variable. The summation notation above, therefore,represents the sum 9 + 16 + 25 + 36 + 49. Thus,we have the index values 3, 4, 5, 6, and 7, and the squares of thoseare 9, 16, 25, 36, and 49. Theindex values begin with 3 and increase by 1 until reaching 7. Thereplace and simplify process continues until the last index value tobe used is the upper limit of summation.ĭetermine the expansion of this summation notation:Įach addend in the sum will be the square of an index value. We repeat this process with the next value of the indexvariable, using that specific value for the index variable in theaddend representation and simplifying as desired or necessary. To expand this summation notation, that is, to determine the setof addends that we are to sum, we replace any occurance of the dummyvariable in the addend representation with the lower limit of theindex variable. Often is is much more involved than this. Here, that expression is just the index variable. The expression to the right of sigma describes or represents each addend, in terms of the index variable i.Here, the largest value i will take on is 10. The upper limit of summation indicates the largest value the index will take on.Unless indicated otherwise, we increase this value by 1 until we reach the upper limit of summation. Here, the smallest value i will take is 1. The lower limit of summation indicates the smallest value the index will take on. ![]() The dummy variable will usually show up one or more times in the expression to the right of the Greek letter sigma. The index of summation, here the letter i, is a dummy variable whose value will change as the addends of the sum change.The Greek letter sigma is closely associated with the word "sum." The letter sigma is a signal that summation notation is being used.The annotated symbolism shown below identifies important elementsused in summation notation (also called sigma notation). For instance, here is the summationnotation to represent the sum of the first 10 positive integers, thefirst sum described on this page. Summation notation provides for us a compact way to represent theaddends in sums such as these. See whether you candetect and describe the addend patterns in the following sums. The smallest addend is 2, each successive addend is 4 larger thanthe previous, and the largest addend is 18. The smallest addend is 1, each successive addend is one largerthan the one before it, and the largest addend is 10. In this content note we discuss and illustrate compact mathematical notation to express certain types of sums and products.Īt times when we add, there is a pattern by which we can expressthe addends. MAT 305: Combinatorics Topics for K-8 Teachers Illinois State University Mathematics Department ![]()
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