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## University of Capital University of Economics and Business Dept' of Computer Science IT222 Applicati

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### ... form formula for the result, discovered by Euler at age 12! ... Leonhard. Euler (1707-1783) 9/27/09. 10. Module #13 - Summations. Euler's Trick, Illustrated ... – PowerPoint PPT presentation

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Title: University of Capital University of Economics and Business Dept' of Computer Science IT222 Applicati

1
University of Capital University of Economics and
IT222Applications of Discrete
StructuresInstructor Xiaoting Zhao
2
Module 13Summations
• Rosen 5th ed., 3.2
• 19 slides, 1 lecture

3
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.

4
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

5
Simple Summation Example

6
More Summation Examples
• An infinite series with a finite sum
• Using a predicate to define a set of elements to
sum over

7
Summation Manipulations
• Some handy identities for summations

(Distributive law.)
(An applicationof commut-ativity.)
(Index shifting.)
8
More Summation Manipulations
• Other identities that are sometimes useful

(Series splitting.)
(Order reversal.)
(Grouping.)
9
• 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!
• And frequently rediscovered by many

LeonhardEuler(1707-1783)
10
Eulers Trick, Illustrated
• Consider the sum12(n/2)((n/2)1)(n-1)n
• We have n/2 pairs of elements, each pair summing
to n1, for a total of (n/2)(n1).

n1

n1
n1
11
Symbolic Derivation of Trick
For case where n is even
12
Concluding Eulers Derivation
• So, you only have to do 1 easy multiplication in
• Also works for odd n (prove this at home).

13
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...

14
Geometric Sum Derivation
• Herewego...

15
Derivation example cont...

16
Concluding long derivation...

17
Nested Summations
• These have the meaning youd expect.
• Note issues of free vs. bound variables, just
like in quantified expressions, integrals, etc.

18
Some Shortcut Expressions

Geometric series.
Eulers trick.
Cubic series.
19
Using the Shortcuts
• Example Evaluate .
• Use series splitting.
• Solve for desiredsummation.