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**Unformatted text preview: **tal Greek letter sigma and is
n=
shorthand for ‘sum’. The index of summation tells us which term to start with and which term
to end with. For example, using the sequence an = 2n − 1 for n ≥ 1, we can write the sum
a3 + a4 + a5 + a6 as
6 (2n − 1) = (2(3) − 1) + (2(4) − 1) + (2(5) − 1) + (2(6) − 1)
n=3 = 5 + 7 + 9 + 11
= 32
The index variable is considered a ‘dummy variable’ in the sense that it may be changed to any
letter without aﬀecting the value of the summation. For instance,
6 6 6 (2k − 1) = (2n − 1) =
n=3 (2j − 1)
j =3 k=3 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
n ak xk f (x) =
k=0 for real numbers ak , k = 0, 1, . . . n. The reader is invited to compare this with what is given in
Deﬁnition 3.1. Summation notation is particularly useful when talking about matrix operations.
For example, we can write the product of the ith row Ri of...

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