1. Discrete Mathematics and its Applications
5/3/2019
University of Florida Dept. of Computer & Information Science & Engineering COT 3100 Applications of Discrete Structures Dr. Michael P. Frank
Slides for a Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen
A word about organization: Since different courses have different lengths of lecture periods, and different instructors go at different paces, rather than dividing the material up into fixed-length lectures, we will divide it up into “modules” which correspond to major topic areas and will generally take 1-3 lectures to cover. Within modules, we have smaller “topics”. Within topics are individual slides.
5/3/2019
(c) , Michael P. Frank
(c) , Michael P. Frank

2. Rosen 5th ed., §3.2 ~19 slides, ~1 lecture
Module #12: Summations
Rosen 5th ed., §3.2
~19 slides, ~1 lecture
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(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

4. Generalized Summations
For an infinite series, we may write:
To sum a function over all members of a set X={x1, x2, …}:
Or, if X={x|P(x)}, we may just write:
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(c) , Michael P. Frank

5. Simple Summation Example
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6. 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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(c) , Michael P. Frank

7. Summation Manipulations
Some handy identities for summations:
(Distributive law.)
(Application of commut- ativity.)
(Index shifting.)
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(c) , Michael P. Frank

8. More Summation Manipulations
Other identities that are sometimes useful:
(Series splitting.)
(Order reversal.)
(Grouping.)
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(c) , Michael P. Frank

9. Example: Impress Your Friends
Boast, “I’m so smart; give me any 2-digit number n, and I’ll 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!
Leonhard Euler ( )
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(c) , Michael P. Frank

10. Euler’s Trick, Illustrated
Consider the sum: 1+2+…+(n/2)+((n/2)+1)+…+(n-1)+n
n/2 pairs of elements, each pair summing to n+1, for a total of (n/2)(n+1).
n+1 …
n+1
n+1
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(c) , Michael P. Frank

11. Symbolic Derivation of Trick
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12. Concluding Euler’s 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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(c) , Michael P. Frank

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...
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(c) , Michael P. Frank

14. Geometric Sum Derivation
Here we go...
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15. Derivation example cont...
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16. Concluding long derivation...
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17. Nested Summations These have the meaning you’d expect.
Note issues of free vs. bound variables, just like in quantified expressions, integrals, etc.
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(c) , Michael P. Frank

18. Some Shortcut Expressions
Geometric series.
Euler’s trick.
Quadratic series.
Cubic series.
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(c) , Michael P. Frank

19. Using the Shortcuts Example: Evaluate . Use series splitting.
Solve for desired summation.
Apply quadratic series rule.
Evaluate.
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(c) , Michael P. Frank

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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(c) , Michael P. Frank