Sequences and Summations in Elementary Discrete Mathematics, Slides of Discrete Mathematics

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Sequences and Summations

Elementary Discrete Mathematics

Sequences

• Consider the function ƒ: N +^ → R where

ƒ( x ) = 1/ x

Then:

• The image of this function can be said to form a sequence:

f (1) = 1 , f (2) =

2 ,^

f (3) =

3 ,^

f (4) =

1,

1

2

,

1

3

,

1

4

,

1

5

,

1

6

,...

Sequences

• The terms of this the previous sequence are:

a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ,...

• Which is:
• Where:

{^ an },^ where^ an =^

n Docsity.com

Sequences

• Again, if the sequence is defined as:
• Then (^1) a 1 =

{^ a (^) n },^ where^ an =^

1

n

2

1 a 2 =

3

1 a 3 =

4

1 a 4 =

5

1 a 5 =

6

1 a 6 =

7

1 a 7 =

and so on...

Summation Notation

• Symbolic way to define the sum of a series.
• Example:

i - index of summation 1 - lower limit of summation 100 - upper limit of summation

100

1

∑ = + +^ +

i =

i

Summation Notation Examples

∑^ =

6

2

3 i

i

∑= +

5

1 (^1 )

k k k

∑^ = =−

m

j m

jm