Block 3 Discrete Systems Lesson 10 –Sequences and Series

Both finite and countable infinite series and much more

one two three four five six seven eight nine ten

1Narrator: Charles EbelingUniversity of Dayton

Summation Notation

1 1

1

1

1 1

...

1 2 ...

n

i m m n ni m

n

i

n

i

n n

i ii i

n n n

i i i ii m i m i m

a a a a a

i n

c nc

ca c a

a b a b

2

Defining Sequences

Sn is the nth term in a sequence that may be finite or infinite

Sn is a function defined on the set of natural numbers Examples:

If Sn = 1/n, then the first 4 terms in the sequence are 1, 1/2, 1/3, 1/4

If Sn = 1/n!, then the first 4 terms in the sequence are 1, 1/2, 1/6, 1/24

The general term for the sequence -1, 4, -9, 16, -25 is Sn = (-1)n n2

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Arithmetic Progression

Sn = a + (n-1)(d) is an arithmetic progression starting at a and incrementing by d

For example, the first six terms of Sn = 3 + (n-1) (4) are 3, 7, 11, 15, 19, 23

4

The limit of an infinite sequence

If for an infinite sequence, s1, s2, …, sn, …

there exists an arbitrarily small > 0 and an m > 0 such that |sn – s| < for all n > m, then s is the limit of the sequence.

lim nns s

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Examples

2 2

2

212 1

lim lim32 3 22

n n

n nn n

n

1 1 1

2 1 2 1 1 1lim 4 4 lim 4 lim 3.5

2 2 2 2

n n

n n nn n n

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Series

The sum of a sequence is called a series. The sum of an infinite sequence is called

an infinite series If the series has a finite sum, then the series

is said to converge; otherwise it diverges A finite sum will always converge

Let Sn = s1 + s2 + … + sn Sn is the sequence of partial sums

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The Arithmetic Series

( ) ( 2 ) ( 3 ) ...

... ( 3) ( 2) [ ( 1) ]nS a a d a d a d

a n d a n d a n d

The sum of the first 100 odd numbers is

1002 99 2 10,000

2nS 10

1 [2 ( 1) ]2 2n n

n nS s s a n d

More to do with arithmetic series

1

2

1

223

1

1

2

1 2 1

6

1

4

n

i

n

i

n

i

n ni

n n ni

n ni

11

For the overachieving student:Prove these results by induction

The Geometric Sequence

2 1

2 3

, , ,...,

examples

2,2 .7 , 2 .7 , 2 .7 ,...2 .7 or 2, 1.4, 0.98, 0.686,...

2, 0.2, 0.02, 0.002; where 2, 0.1, 4

n

n

a ar ar ar

a r n

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The Geometric Series

Sn = a +ar + ar2 + … + arn-1

r Sn = ar + ar2 + … + arn-1 + arn

Sn - r Sn = (1-r) Sn = a - arn

1

1

1lim , 1

1 1

n

n

n

nn

a rS

r

a r aS if r

r r

This is a most

important series.

13

The Geometric Series in Action Find the sum of the following series:

2 3

0

0

1 1 1 1 110 1 ... ... 10

3 3 3 3 3

10 3015

1 1/ 3 2

, 11

n n

n

n

n

aar if r

r

14

15

Future Value of an AnnuityThe are n annual payments of R (dollars) where

the annual interest rate is r. Let S = the future sum after n payments, then

S = R + R(1+r) + R(1+r)2 + … + R(1+r)n-1

1 1

0 0

(1 ) (1 )n n

i i

i i

S r R R r

0 1 2 … n-2 n-1 n

0 R R … R R R

R(1+r)n-1

R(1+r)R(1+r)2

16

More Future Value of an Annuity

1 1

0 0

1 (1 ) (1 ) 1(1 ) (1 )

1 1

n nn ni i

i i

r rS r R R r R R

r r

the sum of a finite geometric series

2 11

... ; 11

n

na r

s a ar ar ar rr

The Binomial Theorem

A really Big Bonus. Isaac Newton’s first great

discovery (1676)

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First, a notational diversion…Factorial notation: n! = 1·2·3 ··· (n-2) ·(n-1) ·nwhere 0! = 1! = 1 and n! = n (n-1)!

( 1)( 2) ( 1) !

1 2 3 ( 1) !( )!

n n n n n r n

r r r r n r

8 8 7 6 5 4 3 2 1 8! 8 7 6

563 1 2 3 5 4 3 2 1 3!(5)! 1 2 3

for example:

!

!( )!

n n n

r n r r n r

A useful fact:18

Here it is…For n integer:

1 2 2 3 3

2 2 1

0

( 1) ( 1)( 2)( )

1 2 1 2 3( 1)

...1 2

n n n n n

n n n

nn r r

r

n n n n na b a na b a b a b

n na b nab b

na b

r

6 6 5 2 3 3 2 4 5 6

6 5 4 2 3 3 2 4 5 6

6 6 6 6 6 6 64

0 1 2 3 4 5 6

6 15 20 15 6

a b a a b a b a b a b ab b

a a b a b a b a b ab b

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Some Observations on the binomial theorem

n+1 terms sum of the exponents in each term is n coefficients equi-distant from ends are equal A related theorem:

1

1

n n n

r r r

This is truly an amazing result!

Look for my triangle on the next slide.

20

Pascal’s triangle

21

1

1

5 4 4

3 2 3

10 6 4

n n n

r r r

The Generalized Binomial Theorem

For n non-integer or negative:

0

( )n n r r

r

na b a b

r

22

Other series worth knowing

binomial series

exponential series

logarithmic series

2 3

0

1 ...2! 3! !

nx

n

x x xe x

n

1 2 3 4 5 2(1 ) 1 ... 1x x x x x x x

2 3 4

ln 1 ... 1 12 3 5

x x xx x x

23

This Series has come to an end

Next Time – Discrete Probability

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