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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
BusinessDept. of Computer Science
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
Example Impress Your Friends
  • 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
    your head, then cut in half.
  • 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.
Quadratic series.
Cubic series.
19
Using the Shortcuts
  • Example Evaluate .
  • Use series splitting.
  • Solve for desiredsummation.
  • Apply quadraticseries rule.
  • Evaluate.

20
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.
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