Module - PowerPoint PPT Presentation

Loading...

PPT – Module PowerPoint presentation | free to download - id: 16505b-Y2RhO



Loading


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation
Title:

Module

Description:

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

Number of Views:32
Avg rating:3.0/5.0
Slides: 22
Provided by: Michael1807
Category:
Tags: euler | leonhard | module

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

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
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!

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
    your head, then cut in half.
  • 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.
Quadratic series.
Cubic series.
18
Using the Shortcuts
  • Example Evaluate .
  • Use series splitting.
  • Solve for desiredsummation.
  • Apply quadraticseries rule.
  • 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
About PowerShow.com