The document outlines lecture notes for a discrete mathematics course taught by Dr. M.M.A. Hashem at Khulna University of Engineering & Technology. It covers topics including sequences, series, summations, products, factorials, and countable sets. Examples of sequences discussed include arithmetic, geometric, and Fibonacci sequences. Formulas are provided for evaluating sums of arithmetic and geometric series.

1. CSE 1107: Discrete Mathematics
Sequence and Sums
M.M.A. Hashem, PhD
Professor
Dept. of Computer Science and Engineering
Khulna University of Engineering & Technology (KUET)
Khulna 9203, Bangladesh
Email: hashem@cse.kuet.ac.bd
Cell Phone: +880 1714 003949
Tuesday, April 9, 2024 CSE 1107: Discrete Mathmatics, CSE, KUET 1

2. Sequence
A sequence is an ordered list of elements.
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3. Examples of Sequence
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4. Examples of Sequence
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5. Examples of Sequence
Not all sequences are arithmetic or geometric sequences.
An example is Fibonacci sequence
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6. Fibonacci Sequence in Nature
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13
8
5
3
2
1

7. Examples of Sequence
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8. More on Fibonacci Sequence
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9. Examples of Golden Ratio
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10. Sequence Formula
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11. Sequence Formula
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12. Some useful sequences
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13. Summation
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14. Evaluating sequences
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15. Arithmetic Series
Consider an arithmetic series a1, a2, a3, …, an. If the common
difference (ai+1 - a1) = d, then we can compute the kth term ak
as follows:
a2 = a1 + d
a3 = a2 + d = a1 +2 d
a4 = a3 + d = a1 + 3d
ak = a1 + (k-1).d
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16. Evaluating sequences
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17. Sum of arithmetic series
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18. Solve this
Calculate 12 + 22 + 32+ 42 + … + n2
[Answer n.(n+1).(2n+1) / 6]
1 + 2 + 3 + … + n = ?
[Answer: n.(n+1) / 2] why?
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19. Can you evaluate this?
Here is the trick. Note that
Does it help?
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20. Double Summation
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21. Sum of geometric series
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22. Sum of infinite geometric series
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23. Solve the following
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24. Sum of harmonic series
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25. Sum of harmonic series
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26. Book stacking example
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27. Book stacking example
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28. Useful summation formulae
See page 157 of Rosen Volume 6
or
See page 166 of Rosen Volume 7
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29. Products
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30. Dealing with Products
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31. Factorial
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32. Factorial
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33. Stirling’s formula
A few steps are omitted here
Here means that the ratio of the two sides approaches 1 as n approaches ∞
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34. Countable sets
Cardinality measures the number of elements in a set.
DEF. Two sets A and B have the same cardinality, if and only if
there is a one-to-one correspondence from A to B.
Can we extend this to infinite sets?
DEF. A set that is either finite or has the same cardinality
as the set of positive integers is called a countable set.
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35. Countable sets
Example. Show that the set of odd positive integers is countable.
f(n) = 2n-1 (n=1 means f(n) = 1, n=2 means f(n) = 3 and so on)
Thus f : Z+  {the set of of odd positive integers}.
So it is a countable set.
The cardinality of an infinite countable set is denoted by
(called aleph null)
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36. Countable sets
Theorem. The set of rational numbers is countable.
Why? (See page 173 of the textbook)
Theorem. The set of real numbers is not countable.
Why? (See page 173-174 of the textbook).
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