James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A34
appendix E Sigma Notation
A convenient way of writing sums uses the Greek letter o (capital sigma, corresponding
to our letter S) and is called sigma notation.
This tells us to
end with i=n.
This tells us
to add.
This tells us to
start with i=m.
n
µ a i
im
1 Definition If a m , a m11 , . . . , a n are real numbers and m and n are integers such
that m < n, then
o n
a i − a m 1 a m11 1 a m12 1 ∙ ∙ ∙ 1 a n21 1 a n
i−m
With function notation, Definition 1 can be written as
o n
f sid − f smd 1 f sm 1 1d 1 f sm 1 2d 1 ∙ ∙ ∙ 1 f sn 2 1d 1 f snd
i−m
Thus the symbol o n i−m indicates a summation in which the letter i (called the index of
summation) takes on consecutive integer values beginning with m and ending with n,
that is, m, m 1 1, . . . , n. Other letters can also be used as the index of summation.
EXAMPLE 1
(a) o 4
i−1
(b) o n
i−3
(c) o 5
j−0
(d) o n
k−1
(e) o 3
i−1
(f) o 4
i−1
i 2 − 1 2 1 2 2 1 3 2 1 4 2 − 30
i − 3 1 4 1 5 1 ∙ ∙ ∙ 1 sn 2 1d 1 n
2 j − 2 0 1 2 1 1 2 2 1 2 3 1 2 4 1 2 5 − 63
1
k − 1 1 1 2 1 1 3 1 ∙ ∙ ∙ 1 1 n
i 2 1
i 2 1 3 − 1 2 1
1 2 1 3 1 2 2 1
2 2 1 3 1 3 2 1
3 2 1 3 − 0 1 1 7 1 1 6 − 13
42
2 − 2 1 2 1 2 1 2 − 8 ■
EXAMPLE 2 Write the sum 2 3 1 3 3 1 ∙ ∙ ∙ 1 n 3 in sigma notation.
SOLUTIOn There is no unique way of writing a sum in sigma notation. We could write
2 3 1 3 3 1 ∙ ∙ ∙ 1 n 3 − o n
i 3
i−2
or
or
n21
2 3 1 3 3 1 ∙ ∙ ∙ 1 n 3 − o s j 1 1d 3
j−1
n22
2 3 1 3 3 1 ∙ ∙ ∙ 1 n 3 − o sk 1 2d 3
The following theorem gives three simple rules for working with sigma notation.
k−0
■
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