arithmetic series additive recursion. 7/15/2013 arithmetic series 2 the art of asking the right...
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Arithmetic Series
Additive Recursion
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The art of asking the right questions in mathematics is more important than the art of solving them
− Georg Cantor
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What is a series ? A sum of a sequence of numbers Definition:
A finite series is a sum of form
a1 + a2 + a3 + ... + an
for some positive integer n NOTE:
A finite series has last term an
Series
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What is a series ? A sum of a sequence of numbers Definition:
An infinite series is a sum of form
a1 + a2 + a3 + • • • + an + • • •
NOTE:
An infinite series has no last term
Series
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nth Partial Sum Sn Sn = a1 + a2 + a3 + ... + an Finite Series
Sum S of all n terms is Sn for some
positive integer n From the closure axiom, Sn exists
and is a number
Partial Sums
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Finite Series Sum of n terms can be written as
Examples:
Summation Notation
∑ akk=1
n
= 2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 30
= 6 + 6 + 6 + 6 = 24
a1 + a2 + a3 + • • • + an =
k=1
5
2k∑
k=1
4
∑ 6
1.
2.
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Arithmetic Series
Partial Sums
Definition:
such that A series ∑ akk=1
n
for some constant d and for all k, k = 1, 2, 3, … , n is an arithmetic series with common difference d
ak+1 = ak + d
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Arithmetic Sums
Partial Sums
= a1 + (a1 + d) + (a1 + 2d) + • • •
+ (a1 + (n – 2)d) + (a1 + (n – 1)d)
Sn = a1 + a2 + a3 + • • • + an–1 + an
For arithmetic series ∑ akk=1
n
= Snwith common
difference d
In reverse order …Sn = (a1 + (n – 1)d) + (a1 + (n – 2)d) + • • •
+ (a1 + 2d) + (a1 + d) + a1
Sn
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Arithmetic Sums
Partial Sums
+ (a1 + (n – 2)d) + (a1 + (n – 1)d)
Adding …
Sn = (a1 + (n – 1)d) + (a1 + (n – 2)d) + • • •
+ • • • (a1 + 2d) + (a1 + d) + a1
= a1 + (a1 + d) + (a1 + 2d) • • • Sn
and …
= [2a1 + (n – 1)d] + [2a1 + (n – 1)d] + • • •
2Sn
n terms
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Arithmetic Sums
Partial Sums
= [2a1 + (n – 1)d] + [2a1 + (n – 1)d] + • • •
2Sn
= n[2a1 + (n – 1)d]
= n[a1 + a1 + (n – 1)d]
an
= n(a1 + an)
Thus
Sn = n(a1 + an )
2
= n • (average of a1 and an )
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ak = area of kth rectangle of unit width
Sn = area of 1st n rectangles
Let n = 5
Then
Arithmetic Series Partial Sums
1 2 3 4 5
a1 a2
a4 a5
Average height
Sn = a1 + an
2 ( )n
= 5 a1 + a5
2 ( )S5
5
a3
a1 + a5
2
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Arithmetic Series
Infinite Series
(2a1 + (n – 1)d)
= 2 n
= na1 +
n(n – 1)
2 d
Recall that Sn n a1 + an
2 ( )=
What happens to Sn as n ∞ ?
What can we say about S
∑ k=1
∞ak=
?
Sn
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Arithmetic Series
Infinite Series
Remember:
The sum of an infinite arithmetic series
never exists !
S does not exist !
What can we say about S
∑ k=1
∞ak=
?
= na1 +
n(n – 1)
2 d
Sn
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Example: Let a1 = -6 with common difference 4
Arithmetic Series
∞ as n ∞
an = a1 + (n – 1)d = -6 + 4(n – 1)
a2 = a1 + 4 = -6 + 4 = -2a3 = a2 + 4 = -2 + 4 = 2a4 = a3 + 4 = 2 + 4 = 6
• • • • • • • • • • • •
= 2n(n – 4 )=Sn
a1 + an
2 ( )n
Thus S does not exist !=k=1
∞ak∑
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Think about it !