Module

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Module 12Summations

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

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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.

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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

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Simple Summation Example

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More Summation Examples

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

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Summation Manipulations

• Some handy identities for summations

(Distributive law.)
(Applicationof commut-ativity.)
(Index shifting.)
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More Summation Manipulations

• Other identities that are sometimes useful

(Series splitting.)
(Order reversal.)
(Grouping.)
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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)
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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
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Symbolic Derivation of Trick
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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).

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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...

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Geometric Sum Derivation

• Herewego...

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Derivation example cont...

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Concluding long derivation...

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Nested Summations

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

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Some Shortcut Expressions

Geometric series.
Eulers trick.
Quadratic series.
Cubic series.
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Using the Shortcuts

• Example Evaluate .
• Use series splitting.
• Solve for desiredsummation.
• Apply quadraticseries rule.
• Evaluate.

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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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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.

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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