Sequences & Series


Arithmetic Mean 
In mathematics and statistics, the arithmetic mean (/ˌærɪθˈmɛtɪk ˈmiːn/), or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection
‎Definition · ‎Motivating properties · ‎Contrast with median · ‎Generalizations

 

Arithmetic Sequence 
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15 … is an arithmetic progression with common difference of 2


 
Arithmetic Series 
noun
noun: arithmetic series; plural noun: arithmetic series
  1. a sequence of numbers in which each differs from the preceding by a constant quantity (e.g., 3, 6, 9, 12, etc.; 9, 7, 5, 3, etc.).
    • the relationship between numbers in an arithmetic progression.

      "the numbers are in arithmetic progression"


Circumscribed Rectangle 

The circumscribed rectangle, or bounding box, is the smallest rectangle that can be drawn around a set of points such that all the points are inside it, or exactly on one of its sides. The four sides of the rectangle are always either vertical or horizontal, parallel to the x or y axis




 
Common Difference 

The difference between each number in an arithmetic sequence



Common Ratio 

For a geometric sequence or geometric series, the common ratio is the ratio of a term to the previous term. This ratio is usually indicated by the variable r. Example: The geometric series 3, 6, 12, 24, 48, . . . has common ratio r = 2




Convergence

Convergence, in mathematics, property (exhibited by certain infinite series and functions) of approaching a limit more and more closely as an argument (variable) of the function increases or decreases or as the number of terms of the series increases. For example, the function y = 1/x converges to zero as x increases






Divergence

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero



Explicit Formula 

Explicit Formula of a Sequence. A formula that allows direct computation of any term for a sequence a1, a2, a3, . . . , an, . . . . 


 
  

Comments

Post a Comment

Popular posts from this blog

Squeeze Theorem - "2 policeman & a drunk"

In calculus, the squeeze theorem (known also as the pinching theorem, the sandwich theorem, the sandwich rule and sometimes the squeeze lemma) is a theorem regarding the limit of a function. The squeeze theorem is a technical result that is very important in proofs in calculus and mathematical analysis. It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Gauss. In Italy, Russia and France, the squeeze theorem is also known as the two carabinieri theorem, two militsioner theorem, two gendarmes theorem, or two policemen and a drunk theorem. The story is that if two policemen are escorting a drunk prisoner between them, and both officers go to a cell, then (regardless of the path taken, and the fact that the prisoner may be wobbling about between the policemen) t...
Read more

Surreal Numbers

Image
In mathematics , the surreal number system is a totally ordered class containing the real numbers as well as infinite and infinitesimal numbers , respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field . [a] If formulated in Von Neumann–Bernays–Gödel set theory , the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions , the Levi-Civita field , the superreal numbers , and the hyperreal numbers , can be realized as subfields of the surreals. [1] It has also been shown (in Von Neumann–Bernays–Gödel set theory ) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice , this need not be the case, and in such the...
Read more