Sigma notation - Mathcentre

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1. Introduction Sigma notation is a concise and convenient way to represent long sums. For example, we often wish to sum a number of terms such as or 1 + 2 + 3 + 4 + 5 1 + 4 + 9 + 16 + 25 + 36 where there is an obvious pattern to the numbers involved. The first of these is the sum of the first five whole numbers, and the second is the sum of the first six square numbers. More generally, if we take a sequence of numbers u 1 , u 2 , u 3 , . . .,u n then we can write the sum of these numbers as u 1 + u 2 + u 3 + . . . + u n . A shorter way of writing this is to let u r represent the general term of the sequence and put n∑ u r . r=1 Here, the symbol Σ is the Greek capital letter Sigma corresponding to our letter ‘S’, and refers to the initial letter of the word ‘Sum’. So this expression means the sum of all the terms u r where r takes the values from 1 to n. We can also write b∑ r=a to mean the sum of all the terms u r where r takes the values from a to b. In such a sum, a is called the lower limit and b the upper limit. u r Key Point The sum u 1 + u 2 + u 3 + . . . + u n is written in sigma notation as n∑ u r . r=1 c○ mathcentre July 18, 2005 2
Exercises 1. Write out what is meant by (a) (e) 5∑ n=1 N∑ n 3 x 2 i i=1 (b) (f) 5∑ 3 n (c) n=1 N∑ f i x i i=1 4∑ (−1) r r 2 r=1 (d) 4∑ k=1 (−1) k+1 2k + 1 2. Evaluate 4∑ k 2 . k=1 2. Some examples Example 4∑ Evaluate r 3 . Solution r=1 This is the sum of all the r 3 terms from r = 1 to r = 4. So we take each value of r, work out r 3 in each case, and add the results. Therefore Example 5∑ Evaluate n 2 . Solution n=2 4∑ r 3 = 1 3 + 2 3 + 3 3 + 4 3 r=1 = 1 + 8 + 27 + 64 = 100 . In this example we have used the letter n to represent the variable in the sum, rather than r. Any letter can be used, and we find the answer in the same way as before: Example 5∑ Evaluate 2 k . k=0 5∑ n 2 = 2 2 + 3 2 + 4 2 + 5 2 n=2 = 4 + 9 + 16 + 25 = 54 . 3 c○ mathcentre July 18, 2005
Page 1: Sigma notation Sigma notation is a
Page 5 and 6: Example 3∑ Evaluate Solution k=1
Page 7 and 8: Suppose we have the sum of a consta
Page 9 and 10: Key Point If a and c are constants,

Sigma notation - Mathcentre

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