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
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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 ∞