Common Sums Discrete Mathematics

9 Sums and Asymptotics Sums and products arise regularly in the analysis of algorithms, nancial applica-tions, physical problems, and probabilistic systems. For example, we have already encountered the sum 1C2C4CC Nwhen counting the number of nodes in a complete binary tree with Ninputs. Although such a sum can be represented

The index is often represented by i. Other common possibilities for representation of the index are j and t. The index appears as the expression i 1. The index assumes values starting with the value on the right hand side of the equation and ending with the value above the summation sign.

Summations are the discrete versions of integrals given a sequence x ax a1x b, its sum x a x a1 x b is written as P b Rosen and Concrete Math-ematics both provide tables of sums in their chapters on generating functions. But it is usually better to be able to reconstruct the solution of a sum rather

All of that will surely make each successive quotpartialquot sum only a very little bit bigger than the previous some, and in some cases, it'll be smaller. Let's compute the partial sums The sum of the first term is just 1. The sum of the first 2 terms is 1 - 13 23 0.666666666666666666

Combinatorics and Discrete Mathematics Applied Discrete Structures Doerr and Levasseur 1 Set Theory 1.5 Summation Notation and Generalizations Sums. Most operations such as addition of numbers are introduced as binary operations. That is, we are taught that two numbers may be added together to give us a single number.

CS 441 Discrete mathematics for CS M. Hauskrecht Arithmetic series Definition The sum of the terms of the arithmetic progression a, ad,a2d, , and is called an arithmetic series. Theorem The sum of the terms of the arithmetic progression a, ad,a2d, , and is Why? 2 1 11 n n S a jd na d j na d n j n j CS 441 Discrete

We can also split a sum up 92sum_i1n a_i 92sum_i15 a_i 92sum_i6n a_i This means that to exclude the first few terms of a sum, we can say 92sum_i6n a_i 92sum_i1n a_i - 92sum_i15 a_i Summations can also be nested 92sum_i1n 92sum_j1n ij

Introduction to Discrete Mathematics Section 3.2 of Rosen email160protected Notes Sequences amp Summations Though you should be at least intuitively familiar with sequences and summations, we give a quick review. Sequences De nition A sequence is a function from a subset of integers to a set S . We use the notations fa ng f a ng1 fan g1 0

initial term common ratio . Remark A geometric progression is a discrete version of the exponential function Examples Write geometric progression with the following parameters a. s s b. t w c. x s u

Sums and products of sequences Sum Summationform Xn km a k a m a m1 a m2 a n where,k index,m lowerlimit,n upperlimit e.g. P n km 1k k1 Product Productform Yn km a k a m a m1 a m2 a n where,k index,m lowerlimit,n upperlimit e.g. Q n km k k1