Aim: What is the summation notation?

Do Now: If an = n2 + n, find the sum of a1 + a2 + a3 + a4 + a5

HW: p.260 # 4,8,10,12,13,16,18,20

The sum a1 + a2 + a3 + a4 + a5 is called a seriesA series is the indicated sum of a sequence:

)55()44()33()22()11( 222225

1

2

iii

We can use the summation notation to present the sum of a sequence (series)

iii

5

1

2 This notation means replace i by 1,2,3,4 and 5, and then add up the resulting values

7030201262

Is read as “the sum from i equals 1 to 5 of i2 + i ”

index of summation lower limit of summation

upper limit of summation

iii

5

1

2

Summation notation (or sigma notation)

The index doesn’t have to be i. Any letter can be used. Also, the index doesn’t have to begin at 1.

Summation notation for an infinite series is similar to that for a finite series. For example, for the infinite series shown earlier, you can write:

302012621

2 iii

The infinity symbol, , indicates that the series continues without end.

Find:

5

0

3k

k 543210 333333 3642438127931

Evaluate:

3

1

2)13(k

k

7

3

2)2(i

iEvaluate:

9364254852 222

52222 )5()4()3()2()1(

552516941

Write the series with summation notation

5 + 10 + 15 + ··· + 100

Notice that the first term is 5 (1), the second is 5 (2),the third is 5 (3), and the last is 5 (20). So the termsof the series can be written as:

ai = 5i where i = 1, 2, 3, . . . , 20

The summation notation is

20

1

5i

i

Write the series with summation notation.

5

4

4

3

3

2

2

1

Notice that for each term the denominator of the fraction is 1 more than the numerator. So, the terms of the series can be written as:

1i

iai Where i = 1,2,3,4…

The summation notation for the series is

1 1i

i

Write the series by sigma notation:

a) 12 + 20 + 30 + 42 + 56 + 72 + 90 + 110

b)

243

5

81

4

27

3

9

2

3

1c)

54321 54321

10

3

2

i

ii

5

1i

ii

5

1 )3(ii

i