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### ... is a simple closed-form formula for the result, discovered by Euler at age 12! Leonhard. Euler ... Concluding Euler's Derivation ... – PowerPoint PPT presentation

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Title: Module

1
Module 12Summations
• Rosen 5th ed., 3.2
• 19 slides, 1 lecture

2
Summation Notation
• Given a series an, an integer lower bound (or
limit) j?0, and an integer upper bound k?j, then
the summation of an from j to k is written and
defined as follows
• Here, i is called the index of summation.

3
Generalized Summations
• For an infinite series, we may write
• To sum a function over all members of a set
Xx1, x2,
• Or, if XxP(x), we may just write

4
Simple Summation Example

5
More Summation Examples
• An infinite series with a finite sum
• Using a predicate to define a set of elements to
sum over

6
Summation Manipulations
• Some handy identities for summations

(Distributive law.)
(Applicationof commut-ativity.)
(Index shifting.)
7
More Summation Manipulations
• Other identities that are sometimes useful

(Series splitting.)
(Order reversal.)
(Grouping.)
8
• Boast, Im so smart give me any 2-digit number
n, and Ill add all the numbers from 1 to n in my
head in just a few seconds.
• I.e., Evaluate the summation
• There is a simple closed-form formula for the
result, discovered by Euler at age 12!

LeonhardEuler(1707-1783)
9
Eulers Trick, Illustrated
• Consider the sum12(n/2)((n/2)1)(n-1)n
• n/2 pairs of elements, each pair summing to n1,
for a total of (n/2)(n1).

n1

n1
n1
10
Symbolic Derivation of Trick
11
Concluding Eulers Derivation
• So, you only have to do 1 easy multiplication in
• Also works for odd n (prove this at home).

12
Example Geometric Progression
• A geometric progression is a series of the form
a, ar, ar2, ar3, , ark, where a,r?R.
• The sum of such a series is given by
• We can reduce this to closed form via clever
manipulation of summations...

13
Geometric Sum Derivation
• Herewego...

14
Derivation example cont...

15
Concluding long derivation...

16
Nested Summations
• These have the meaning youd expect.
• Note issues of free vs. bound variables, just
like in quantified expressions, integrals, etc.

17
Some Shortcut Expressions

Geometric series.
Eulers trick.
Cubic series.
18
Using the Shortcuts
• Example Evaluate .
• Use series splitting.
• Solve for desiredsummation.
• Evaluate.

19
Summations Conclusion
• You need to know
• How to read, write evaluate summation
expressions like
• Summation manipulation laws we covered.
• Shortcut closed-form formulas, how to use them.

20
Cardinality
• Definition 4 Sets A and B have the same
cardinality if and only if there is a one-to-one
correspondence from A to B.
• A set that is either finite or has the same
cardinality as the set of positive integers is
called countable.

21
Cardinality-continued
• Which one is countable? And Why?
• Odd positive integers?
• Positive rational numbers?
• Real numbers?
• Example 19 set of positive rational number is
countable.
• Example 20 set of real numbers is uncountable