Concrete Mathematics: Exploring Sums

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1 Concrete Mathematics: Exploring Sums Anmol Sakarda Nhat Pham May 20, 2018 Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

2 Outline 1 Abstract 2 Sums and Recurrences 3 Manipulating Sums 4 Multiple Sums 5 Finite and Infinite Calculus 6 Infinite Series Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

3 Abstract In this paper, we explore various methodologies to calculate sums. We begin by looking at simple methods: converting sums into recurrences. We proceed to discuss common manipulations, such as using the repertoire method or perturbation. Finally, we conclude by extrapolating sums to both finite and infinite calculus mainly by building integrals. Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

4 Sums and Recurrences The sum is the recurrence S n = n k=0 S 0 = a 0 a k S n 1 + a n for n > 0 Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

5 Sums and Recurrences Continued Therefore, sums can be evaluated in closed form by solving recurrences. Assuming a n equals a constant plus a multiple of n, the recurrence above can be rewritten as R 0 = α R n = R n 1 + β + γn for n > 0 Thus, we can generalize a solution such that: R(n) = A(n)α + B(n)β + C(n)γ Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

6 Manipulating Sums Distributive Property ca k = c k K k K a k Associative Property k K(a k + b k ) = k K a k + k K b k Commutative Property (a k ) = k K p(k) K a p(k) Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

7 Manipulating Sums Continued Iverson Convention From any logical statements made in middle of a formula, the values 0 or 1 can be obtained. Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

8 Multiple Sums Continued Example The following example demonstrates an important rule for combining different sets of indices. If K and K are any sets of integers, then a k + a k = a k + a k k K k K k K K k K K This is derived from the general formulas a k = a k [k K] k K k and [k K] + [k K ] = [k K K ] + [k K K ] We use this method to either combine 2 almost disjoint index sets or to split off a single term from a set. Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

9 Manipulating Sums Continued Perturbation Start with an unknown sum and call it S n + a n+1 = a k = a k n+1 1 k n+1 = a k+1 n+1 = a k n a k+1 a k+1 This expression can then be simply expressed in terms of S n a k Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

10 Manipulation Sums Perturbation Example We then have S n = k2 k 0 k n S n + (n + 1)2 n+1 = (k + 1)2 (k+1) 0 k n Using the associative property gives k2 (k+1) + 0 k n 0 k n 2 (k+1) Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

11 Manipulation Sums Perturbation Example The first sum is simply 2S n The second sum is just a geometric progression that yields 2 n+2 2 Therefore, the total sum becomes (n 1)2 n Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

12 Multiple Sums Multiple Sums The terms of a sum might be specified by two or more indices. General Distributive Law j J,k K a j b k = ( j J a j ) + ( k K b k ) Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

13 Multiple Sums Continued Example Evaluate Replace k with k + j S n = = Summing on j, which is trivial, gives = 1 k n 1 j<k n 1 j<k+j n 1 k j 1 k 1 j n k 1 k = 1 k n n k k Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

14 Multiple Sums Continued Example Then, by the associative law, we have 1 k n n k 1 k n 1 = n( 1 k n 1 k ) n This has the harmonic mean, and is a final answer for the original sum. Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

15 Finite Calculus Difference Operator, Definite Sum, and Shift f (x) = f (x + 1) f (x) b g(x)δx = g(k) a a k<b Ef (x) = f (x + 1) Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

16 Finite Calculus Properties Multiplicative Rule: fg = f g + Eg f Integration analogue b g(x)δx = a a g(x)δx b b g(x)δx + a c g(x)δx = b c g(x)δx a Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

17 Finite Calculus Properties (cont.) Fundamental Theorem of (Finite) Calculus: g(x) = f (x) b g(x) = f (b) f (a) a Chain Rule: u v = uv Ev u Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

18 Finite Calculus Analogues: Rising and Falling Powers Definition Rising and Falling Powers m > 0 : x m = x(x 1)(x 2)...(x m + 1) m > 0 : x m = x(x + 1)(x + 2)...(x + m 1) x 0 = x 0 = 1 m > 0 : x m 1 = (x + 1)(x + 2)...(x + m) m > 0 : x m = 1 (x 1)(x 2)...(x m) Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

19 Finite Calculus Analogues: Rising and Falling Powers Properties of Rising and Falling Powers x m = mx m 1 k m = nm+1 m k<n a k<b c k = cb c a c 1 Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

20 Infinite Series Definition Infinite Series: Suppose a k 0 k K. If there is a number A such that for any finite F K: a k A k F Then k K a k exists and is equal to the least such A. Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

21 Infinite Series Generalization for any (multi-dimensional) index set K and any sequence a k : A = a k = a + k a k k K k K k K where: a + k = a k.[a k > 0] a k = a k.[a k < 0] Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

22 Infinite Series We have the following scenarios: A + and A are finite: A converges absolutely to A + + A. A + is infinite, A is finite: A diverges to. A is infinite, A + is finite: then A diverges to. A + and A are infinite: A is undefined. Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

23 Infinite Series Fundamental Principle of Multiple Sums Absolutely convergent sums over 2 or more indices can always be summed first with respect to any one of those indices. Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

24 Infinite Series Formally, if J and the elements of {K j j J} are sets of indices such that: a j,k converges absolutely to A j J,k K j Then there exists A j for each j J such that a j,k converges absolutely to A j k K j A j converges absolutely to A j J Anmol Sakarda, Nhat Pham Concrete Mathematics: Exploring Sums May 20, / 24

Sums. ITT9131 Konkreetne Matemaatika

Sums. ITT9131 Konkreetne Matemaatika Sums ITT9131 Konkreetne Matemaatika Chapter Two Notation Sums and Recurrences Manipulation of Sums Multiple Sums General Methods Finite and Innite Calculus Innite Sums Contents 1 Sequences 2 Notations