Infinite Sides
Infinite Sides - Related Videos
01:28
Introduction to Infinite Series
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2026
An infinite series is the sum of an infinite sequence of terms. Instead of adding only a fixed number of values, the addition continues without end. To make sense of this process, mathematicians examine partial sums, which are running totals formed by adding the first few terms of the series. If these partial sums approach a fixed number, the infinite series is said to converge. If they do not approach a finite value, the series diverges.The water tank example illustrates convergence through...
01:28
Infinitely Repeated Games
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2025
An infinitely repeated game is a scenario where players repeatedly engage in the same game without a predetermined end. This concept is crucial in understanding long-term interactions in various fields of the social sciences, including economics and international relations. Examples: Firms Setting Prices: Companies continuously adjust prices. Trade Negotiations: Countries consistently negotiate trade terms. Tit-for-Tat Strategy Cooperative Start: Players begin with a cooperative action,...
07:28
A Contusive Model of Unilateral Cervical Spinal Cord Injury Using the Infinite Horizon Impactor
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Cited by 46 •
2012
A reliable and repeatable way to produce a cervical unilateral spinal cord injury using the Infinite Horizon impactor is described. The method takes advantage of a custom designed frame and clamp to stabilize the spine. The standardized procedure and biomechanical injury parameters result in sufficient and sustained injuries.
01:29
Improper Integrals: Infinite Intervals
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2026
An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.Application to Exponential...
02:35
Reaction Quotient
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2020
The status of a reversible reaction is conveniently assessed by evaluating its reaction quotient (Q). For a reversible reaction described by m A + n B ⇌ x C + y D, the reaction quotient is derived directly from the stoichiometry of the balanced equation as where the subscript c denotes the use of molar concentrations in the expression. If the reactants and products are gaseous, a reaction quotient may be similarly derived using partial pressures: Note that the reaction quotient equations...
FAQs
Each additional side divides the total turning into smaller increments. For a regular construction, the exterior angle becomes progressively smaller as the side count increases, while the direction changes more gradually from one edge to the next. This shrinking angular step explains why the boundary begins to resemble a smooth curve rather than a visibly angular figure.
When regular polygons remain inscribed in a fixed circle, adding sides places their vertices and edges closer to the circular boundary. The resulting polygonal boundaries converge toward the circle as the side count grows without bound. This provides a geometric example of how a continuous curve can arise as the limiting form of increasingly fine discrete approximations.
Increasing the number of sides makes each individual edge shorter, approaching zero in the limiting process. However, the growing number of edges compensates for that decrease, so the total perimeter approaches the fixed circle’s circumference. This contrast illustrates why the behavior of individual parts cannot always be inferred from the behavior of their sum.
Begin with a regular polygon inscribed in a fixed circle, then increase its number of equal sides while keeping the same circle. Compare the polygon’s boundary and perimeter after each increase. The edges become shorter, the shape follows the circle more closely, and the perimeter approaches the circle’s circumference, making the construction a practical limit-based approximation.
The construction supplies an intuitive model for a limit: a sequence of discrete figures changes as the side count grows without bound, and its measurable features approach those of a continuous curve. Studying shrinking exterior angles, vanishing side lengths, and converging perimeters helps connect geometric reasoning with the limiting processes used throughout calculus.
A curved boundary can be represented by a sequence of increasingly refined polygonal approximations. By increasing the number of sides, numerical work can produce perimeter estimates that move closer to the circumference of the fixed circle. The method demonstrates how discrete calculations can approximate continuous geometric quantities and provides a foundation for analyzing approximation quality through successive refinements.