Lecture 10: Sequences and Summations (2) Dr. Chengjiang Long - PowerPoint PPT Presentation

Lecture 10: Sequences and Summations (2) Dr. Chengjiang Long Computer Vision Researcher at Kitware Inc. Adjunct Professor at SUNY at Albany. Email: clong2@albany.edu

Outline Special sequences • Sum of the elements of a sequence • 2 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Outline Special sequences • Sum of the elements of a sequence • 3 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Sequences Definition : A sequence is a function from a subset of • integers to a set S. We use the notation(s): & { a n } #$% {a n } or Each a n is called the n th term of the sequence • We rely on the context to distinguish between a • sequence and a set, although they are distinct structures 4 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Geometric Sequence • Definition : A geometric progression is a sequence of the form a, aq, aq 2 , aq 3 , …, aq n , … Where: a Î R is called the initial term • q Î R is called the common ratio • • A geometric progression is a discrete analogue of the exponential function f(x) = aq x 5 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Arithmetic Sequence • Definition : An arithmetric progression is a sequence of the form a, a+d, a+2d, a+3d, …, a+nd, … Where: a Î R is called the initial term • d Î R is called the common difference • • An arithmetic progression is a discrete analogue of the linear function f(x) = dx+a 6 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

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 a 3 = a 2 + d = a 1 +2 d a 4 = a 3 + d = a 1 + 3d a k = a 1 + (k-1).d 7 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Fibonacci Sequence Golden ratio. 8 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Harmonic Sequence ( The sequence: {ℎ # } %&' = 1/n • is known as the harmonic sequence The sequence is simply: • 1, 1/2, 1/3, 1/4, 1/5, … This sequence is particularly interesting because its • summation is divergent: ( ∑ %&' (1/n) = ¥ 9 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Some useful sequences 10 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Outline Special sequences • Sum of the elements of a sequence • 11 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Summation You should be by now familiar with the summation • notation: % ∑ "#$ & " = a m + a m+1 + … + a n-1 + a n Here i is the index of the summation • m is the lower limit • n is the upper limit • Often times, it is useful to change the lower/upper • limits, which can be done in a straightforward manner (although we must be very careful): %)' & "*' % ∑ "#' & " = ∑ "#( 12 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Summation 13 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Summation of Geometric Sequence With 5 terms of the general geometric sequence, we have = + + + + 2 3 4 S a ar ar ar ar 5 TRICK Multiply by r: = + 2 + 3 + 4 + 5 rS ar ar ar ar ar 5 Subtracting the expressions gives - = + + + + 2 3 4 S rS a ar ar ar ar 5 5 - + + + + 2 3 4 5 ar ar ar ar ar Move the lower row 1 place to the right and subtract 14 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Summation of Geometric Sequence With 5 terms of the general geometric sequence, we have = + + + + 2 3 4 S a ar ar ar ar 5 TRICK Multiply by r: = + 2 + 3 + 4 + 5 rS ar ar ar ar ar 5 Subtracting the expressions gives - = + + + + 2 3 4 S rS a ar ar ar ar 5 5 - + + + + 2 3 4 5 ar ar ar ar ar - = - 5 S rS a ar 5 5 15 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Summation of Geometric Sequence - = - 5 S rS a ar So, 5 5 Take out the common factors 5 - = - S 5 a ( 1 r ) ( 1 r ) and divide by ( 1 – r ) 5 - a ( 1 r ) Þ = S 5 - 1 r Similarly, for n terms we get n - a ( 1 r ) = S n - 1 r 16 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Summation of Geometric Sequence The formula n - a ( 1 r ) = S n - 1 r gives a negative denominator if r > 1 Instead, we can use n - a ( r 1 ) = S n - r 1 17 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Example Find the sum of the first 20 terms of the geometric series, - + - + 2 6 18 54 . . . leaving your answer in index form - 3 - 6 = = = - Solution : a 2 , r 3 2 1 ( ) ( ) n - 20 a ( 1 r ) - - 2 1 3 = S Þ = S ( ) n 20 - - - 1 r 1 3 We’ll simplify this answer without using a calculator 18 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Example ( ) ( ) 20 - - 2 1 3 Þ = S ( ) 20 - - 1 3 There are 20 minus signs here and 1 more outside the bracket! ( ) 1 20 - 2 1 3 = 4 2 20 - 1 3 = 2 19 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Sum of arithmetic series What is the sum of this series? 50 å ( ) - 73 2 p = p 1 ( ) ( ) = + + + + - + - 71 69 67 . . . 25 27 Write the first three terms and the last two terms of the • following arithmetic series. 20 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Sum of arithmetic series What is the sum of these terms? ( ) ( ) + + + + - + - 71 69 67 . . . 25 27 ( ) ( ) - + - + + + + 27 25 . . . 67 69 71 + + + + + + 44 44 44 . . . 44 44 44 50 Terms ( ) ( ) + - 50 71 27 = 71 + (-27) Each sum 2 is the same. = 1100 21 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Sum of arithmetic series In general ( ) ( ) ( ) ( ) + + + + + + + - a a d a 2 d . . . a n 1 d 1 1 1 1 ( ) ( ) ( ) ( ) + - + + + + + + a n 1 d . . . a 2 d a d a 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) + é + - ù + + é + - ù + + + é + - ù a a n 1 d a a n 1 d . . . a a n 1 d ë û ë û ë û 1 1 1 1 1 1 s = ì S Sum ( ) + n a a ï = n Number of Terms ï = 1 n í = a First Term 2 ï 1 ï = a Last Term î n 22 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Example ( ) + + + + + - 151 147 143 139 . . . 5 What term is -5? ( ) ( ) = + - a a n 1 d + n a a n 1 = 1 n S ( )( ) - = + - - 5 151 n 1 4 2 = n 40 = n 40 ( ) + - 40 151 5 = S = a 151 2 1 = = - S 2920 a 5 40 23 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Sum of arithmetic series Alternate formula for the sum of an Arithmetic Series. ( ) = + - Substitute a a n 1 d n 1 ( ) + n a a = 1 n S 2 ( ) ( ) + + - = ì n a a n 1 d # of Terms n 1 1 = S ï 2 = 1st Term a í ( ) 1 ( ) + - n 2 a n 1 d ï = d Difference 1 = S î 2 24 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Solve this 36 å ( ) + = + + + + 2.25 0.75 j 2.25 3 3.73 4.5 . . . = j 0 ( ) It is not convenient to find ( ) + - n 2 a n 1 d the last term. 1 = S 2 ( ) = ( ) ( )( ) n 37 + - 37 2 2.25 37 1 0.75 = S = a 2.25 2 1 = = S 582.75 d 0.75 25 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Evaluating sequences 26 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Series • When we take the sum of a sequence, we get a series • We have already seen a closed form for geometric series • Some other useful closed forms include the following: % ∑ "#$ 1 = u-k+1, for k £ u • % ∑ "#$ ' = n(n+1)/2 • ' * = n(n+1)(2n+1)/6 ) ∑ "#( • ' $ » n k+1 /(k+1) ) ∑ "#( • 27 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Infinite Series Although we will mostly deal with finite series (i.e., an • upper limit of n for fixed integer), inifinite series are also useful Consider the following geometric series: • S n=0 (1/2 n ) = 1 + 1/2 + 1/4 + 1/8 + … converges to 2 • S n=0 (2 n ) = 1 + 2 + 4 + 8 + … does not converge • However note: S n=0 (2 n ) = 2 n+1 – 1 (a=1, q=2) • 28 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Evaluating sequences 29 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Can you evaluate this? Here is the trick. Note that Does it help? 30 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Double Summation 31 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018

Solve the following 32 C. Long ICEN/ICSI210 Discrete Structures Lecture 10 September 25, 2018