Math 111 - 11/28/11

8.1 Intro to sequences, series, summation notation...

A "Sequence" is a set of numbers in some specific order

Ex:

{1,2,3,4,5,...}

{2,4,6,8,10,...}

{5,10,15,20,...}

{0,0,0,0,0,...}

{1,11,21,1211,543,8,-7...}

A sequence can be infinite (like all of the above)

or finite

Ex

{1}

{5,3,1}

{2,2,222}

And no specific pattern is required (but there usually is one anyway).

The numbers in a sequence are called its "terms" and we use subscripts to refer to them...

Learn Math Equation: a =({ 2 , 4 , 6 , 8 , 1 0 ,s.)}

Learn Math Equation: a(_ 2)_ = 4

Learn Math Equation: a(_ 5)_ = 1 0

Learn Math Equation: a(_ 1 0)_ = 2 0

b = {1,11,21,1211,543,8,-7...}

Learn Math Equation: b(_ 1)_ = 1

Learn Math Equation: b(_ 3)_ = 2 1

If a sequence has the right pattern, you can a write a formula for the terms...

For example Learn Math Equation: a =({ 2 , 4 , 6 , 8 , 1 0 ,s.)}

Has the formula Learn Math Equation: a(_ n)_ = 2 n

{1,2,3,4,5,...} Learn Math Equation: a(_ n)_ = n

{5,10,15,20,...} Learn Math Equation: a(_ n)_ = 5 n

{0,0,0,0,0,...}Learn Math Equation: a(_ n)_ = 0

So I could tell you Learn Math Equation: b(_ n)_ = 3 n - 7

Learn Math Equation: b =({ - 4 , - 1 , 2 , 5 ,s.)}

Learn Math Equation: b(_ 1 0 0)_ = 3s* 1 0 0 - 7 = 2 9 3

Learn Math Equation: d =({(/ 2,/ 3)/ ,(/ 3,/ 4)/ ,(/ 4,/ 5)/ ,(/ 5,/ 6)/ ,(/ 6,/ 7)/ ,s.)}

Learn Math Equation: d(_ 3)_ =(/ 4,/ 5)/

Learn Math Equation: d(_ 1 0)_ =(/ 1 1,/ 1 2)/

Learn Math Equation: d(_ n)_ =(/ n + 1,/ n + 2)/

Learn Math Equation: g =({ 1 , 4 , 9 , 1 6 , 2 5 , 3 6 ,s.)}

Learn Math Equation: g(_ n)_ = n(^ 2)^

A "series" is a sum of the beginning of a sequence...

Ex: Learn Math Equation: a =({ 2 , 4 , 6 , 8 , 1 0 ,s.)}

The "first partial sum" is Learn Math Equation: s(_ 1)_ = 2

The "second partial sum" is Learn Math Equation: s(_ 2)_ = 2 + 4 = 6

The "third partial sum" is Learn Math Equation: s(_ 3)_ = 2 + 4 + 6 = 1 2

The 6th partial sum is Learn Math Equation: s(_ 6)_ = 2 + 4 + 6 + 8 + 1 0 + 1 2 = 4 2

These make a sequence of their own... Learn Math Equation: s =({ 2 , 6 , 1 2 , 2 0 , 3 0 , 4 2 ,s.)} this is a series (the sums of a sequence)

Summation notation:

We use a greek capitol sigma to represent sums...

Ex: Learn Math Equation:(s n = 1,s 5)s 2 n

means Learn Math Equation: 2s* 1 + 2s* 2 + 2s* 3 + 2s* 4 + 2s* 5

Ex: Learn Math Equation:(s n = 3,s 7)s(/ 1,/ n)/ =(/ 1,/ 3)/ +(/ 1,/ 4)/ +(/ 1,/ 5)/ +(/ 1,/ 6)/ +(/ 1,/ 7)/ = ?

Ex: Learn Math Equation:(s n = 1 0,s 2 0)s 5 = 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = ?

Ex: Learn Math Equation:(s n = 2,s 5)s(( - 1))(^ n)^ =(( - 1))(^ 2)^ +(( - 1))(^ 3)^ +(( - 1))(^ 4)^ +(( - 1))(^ 5)^ = 0

Series vs. sequence...

Learn Math Equation: a(_ n)_ =(/ n + 1,/ n + 2)/ =({(/ 2,/ 3)/ ,(/ 3,/ 4)/ ,(/ 4,/ 5)/ ,(/ 5,/ 6)/ ,s.)} is a sequence

Learn Math Equation: s(_ p)_ =(s n = 1,s p)s(/ n + 1,/ n + 2)/ are the partial sums of that sequence

so

Learn Math Equation: a(_ 5)_ =(/ 5 + 1,/ 5 + 2)/ =(/ 6,/ 7)/

but

Learn Math Equation: s(_ 5)_ =(/ 2,/ 3)/ +(/ 3,/ 4)/ +(/ 4,/ 5)/ +(/ 5,/ 6)/ +(/ 6,/ 7)/ = ?