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Summation NotationSummation NotationSummation NotationSummation Notation

Math 115A Spring 2008

Dawson

DefinitionsDefinitions^

Summation notation is used to write asum of multiple terms efficiently^ The Greek letter capital sigma,

,is

6 5 4 3 2 1

6 1

x x x x x x x i

i^

∑=

^

The Greek letter capital sigma,

,is

called the summation operator

^

Here the letter

i

is the index of

summation, which tells you the variable inthe expression to sum over

ExamplesExamples

3

3

3

4 1

3

3

∑= i

i

30

12 9 6 3

4 3 3 3 2 3 1 3

) 1 (^5) ( 3 ) 1 (^4) ( 3 ) 1 (^3) ( 3 ) 1 (^2) ( 3 ) 1 ( 3 5 2

=

=

⋅ + ⋅ + ⋅ + ⋅ =

− + − + − + − = −

∑= n

n

PropertiesProperties^

Property 1
Example

∑

∑

∑

=^

=

=

±

=

±

n i

n i

i

i

n i

i i^

b

a

b

a^

1

1

1

3

2

1

3 1

∑= k

k k

3 2 1

3 1

3 1

1

• • • • • = ∑

∑

=^

=

= k

k

k

k

k

PropertiesProperties^

Property 3Suppose

∑

∑

∑

=

n

i

m

i

n

i^

a

a

a

n m

^

Notice on the right hand side, the firstlower limit and the second upper limitmatch the limits on the left hand side

∑

∑

∑

=

=

=^

m i

i

i

i

i

i

1

1

1

PropertiesProperties^

Re-indexing

∑

∑

• +=

−

=

=

c n

c j

c j

n i

i^

a

a^

1

1

^

Notice that the upper and lower limitsincrease (or decrease) by the sameamount, while the index of summationdecreases (or increases) by that amount