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Lecture

Summation and Sequences: Notation and Formulas

In course

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics as diverse as mathematical reasoning, combinatorics, disc ...

Description

This lecture covers summation notation for sequences, including the sum of terms and the index of summation. It also introduces product notation and explores geometric series, with important formulas and closed forms.

Instructor

Karl Aberer

Karl Aberer received his PhD in mathematics in 1991 from the ETH Zürich. From 1991 to 1992 he was postdoctoral fellow at the International Computer Science Institute (ICSI) at the University of California, Berkeley. In 1992, he joined the Integrated Publication and Information Systems institute (IPSI) of GMD in Germany, where he was leading the research division Open Adaptive Information Management Systems. In 2000 he joined EPFL as full professor. Since 2005 he is the director of the Swiss National Research Center for Mobile Information and Communication Systems ( NCCR-MICS, www.mics.ch ). He is member of the editorial boards of VLDB Journal, ACM Transaction on Autonomous and Adaptive Systems and World Wide Web Journal. He has been consulting for the Swiss government in research and science policy as a member of the Swiss Research and Technology Council ( SWTR ) from 2003 - 2011.

Official source

Related concepts (19)

Summation

In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined. Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.

Einstein notation

In mathematics, especially the usage of linear algebra in mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.

Abstract index notation

Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus.

Penrose graphical notation

In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A diagram in the notation consists of several shapes linked together by lines. The notation widely appears in modern quantum theory, particularly in matrix product states and quantum circuits. In particular, Categorical quantum mechanics which includes ZX-calculus is a fully comprehensive reformulation of quantum theory in terms of Penrose diagrams, and is now widely used in quantum industry.

Geometric series

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series is geometric, because each successive term can be obtained by multiplying the previous term by . In general, a geometric series is written as , where is the coefficient of each term and is the common ratio between adjacent terms.

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