/
CS 2210:0001 Discrete Structures CS 2210:0001 Discrete Structures

CS 2210:0001 Discrete Structures - PowerPoint Presentation

conchita-marotz
conchita-marotz . @conchita-marotz
Follow
349 views
Uploaded On 2019-12-08

CS 2210:0001 Discrete Structures - PPT Presentation

CS 22100001 Discrete Structures Sequence and Sums Fall 2019 Sukumar Ghosh Sequence A sequence is an ordered list of elements Examples of Sequence Examples of Sequence Examples of Sequence Not all sequences are arithmetic or geometric sequences ID: 769637

series sequence sequences sum sequence series sum sequences examples approaches geometric arithmetic means formula summation harmonic book factorial evaluating

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "CS 2210:0001 Discrete Structures" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

CS 2210:0001 Discrete StructuresSequence and Sums Fall 2019 Sukumar Ghosh

Sequence A sequence is an ordered list of elements.

Examples of Sequence

Examples of Sequence

Examples of Sequence Not all sequences are arithmetic or geometric sequences. An example is Fibonacci sequence

Examples of Sequence

More on Fibonacci Sequence

Examples of Golden Ratio

Sequence Formula

Sequence Formula

Some useful sequences

Summation   Upper limit of j Lower limit of j It means + + + … + +  

Evaluating sequences

Arithmetic Series Consider an arithmetic series a 1 , a 2 , a 3 , …, a n. If the common difference (a i+1 - a 1 ) = d , then we can compute the k th term a k as follows: a 2 = a 1 + d a3 = a2 + d = a1 +2 d a4 = a 3 + d = a1 + 3d ak = a1 + (k-1).d

Evaluating sequences

Sum of arithmetic series

Solve this Calculate 1 2 + 2 2 + 3 2 + 4 2 + … + n 2 [Answer n.(n+1).(2n+1) / 6 ] 1 + 2 + 3 + … + n = ? [Answer: n.(n+1) / 2 ] why?

Can you evaluate this? Here is the trick. Note that Does it help?

Double Summation

Sum of geometric series Thus , So,

Sum of infinite geometric series So,

Sum of infinite geometric series What is the total vertical distance covered by the ball?

Solve the following 0.9 + 0.99 + 0.999 + 0.9999 + 0.99999 + ….. = ?

Sum of harmonic series

Sum of harmonic series If n approaches infinity, then H n approaches infinity, which means the sum of a harmonic series diverges.

Book stacking example

Book stacking example Lesson: You can build an arbitrarily large overhang

Useful summation formulae See page 176 of Rosen Volume 8

Products

Dealing with Products

Factorial

Factorial

Stirling’s formula A few steps are omitted here Here ~ means that the ratio of the two sides approaches 1 as n approaches ∞