Name: infinite geometric series and decay
Uploaded: 2025-11-26T05:43:52.000Z
Duration: 1 min 30 s
Channel: JoVE

Considere una pelota que se deja caer desde cierta altura sobre una superficie dura. Después de cada rebote, rebota a una fracción fija de la altura que alcanzó en el rebote anterior.Las alturas de rebote siguen un patrón geométrico, y cada nueva altura es una fracción constante de la anterior.El valor fijo utilizado para la multiplicación se llama razón común.El primer término representa la altura de rebote inicial, y cada altura posterior se calcula multiplicando el término anterior por la razón común.Este proceso modela la decadencia exponencial, ya que la pelota pierde energía con cada rebote.Posteriormente, la distancia total acumulada recorrida después de varios rebotes se calcula duplicando cada altura de rebote para tener en cuenta tanto el ascenso como el descenso, excepto la caída inicial.Una vez que se duplica cada término de la secuencia, se pueden usar sumas parciales, definidas como la suma de un número fijo de términos, para calcular la distancia acumulada total de la pelota recorrida en cualquier punto.Para encontrar la enésima suma parcial, escribe la suma y multiplícala por la proporción, desplazando todos los términos hacia adelante. Restar cancela los términos medios, dejando solo el primero y el último.La factorización y la división producen la enésima suma parcial, que muestra cómo funcionan las secuencias geométricas en situaciones reales.

sucesiones geométricas

Geometric Sequences

En sistemas donde los valores disminuyen en una proporci&#243;n constante en cada etapa, la sucesi&#243;n resultante sigue una estructura geom&#233;trica. Cada nuevo valor de la sucesi&#243;n se obtiene aplicando un multiplicador fijo al t&#233;rmino anterior. Este tipo de disminuci&#243;n regular y proporcional se utiliza a menudo para representar procesos con p&#233;rdida gradual, como la disipaci&#243;n de energ&#237;a o la reducci&#243;n de la amplitud a lo largo del tiempo.Al analizar el efecto total de un proceso de este tipo a lo largo de infinitas iteraciones, la serie de valores se denomina serie geom&#233;trica infinita. Si la magnitud del multiplicador com&#250;n (la raz&#243;n com&#250;n) es menor que uno, los valores de la sucesi&#243;n contin&#250;an disminuyendo y su total acumulado se aproxima a un valor finito. Esta condici&#243;n garantiza la convergencia, en lugar de un crecimiento o una fluctuaci&#243;n ilimitados.La f&#243;rmula para determinar la suma de una serie geom&#233;trica infinita es:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:3.6%;'><img style='aspect-ratio:289/153;' src='https://cloudfront.jove.com/files/ftp_upload/40095/40095_Equation_1_editor_40095.svg' alt='40095_Equation_1.svg' /></figure>donde S representa la suma de la serie, a es el valor inicial y r es la raz&#243;n com&#250;n. Esta expresi&#243;n matem&#225;tica permite calcular de forma eficiente los resultados a largo plazo en procesos que presentan decrecimiento exponencial. Sus aplicaciones abarcan diversos &#225;mbitos, como la f&#237;sica, la econom&#237;a y la ingenier&#237;a, donde estos modelos de decaimiento exponencial son esenciales para una predicci&#243;n y un an&#225;lisis precisos.

Sucesiones geométricas

En sistemas donde los valores disminuyen en una proporci&#243;n constante en cada etapa, la sucesi&#243;n resultante sigue una estructura geom&#233;trica. ...

Education

Educación

JoVE Core

Math Fundamentals

Fundamentos de matemáticas

Unit

Unidad

Introduction to Sequences and Series

Introducción a las sucesiones y series

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

infinite geometric series and decay

Infinite geometric series describe values that shrink by the same factor each step. Each new term is found by multiplying the previous term by a fixed common ratio. This pattern is useful for modeling gradual loss, such as energy dissipation or a drop in amplitude over time.

When the common ratio has a magnitude less than one, the terms keep getting smaller. In that case, the total of the series approaches a definite number instead of growing without bound or changing back and forth. This is called convergence. The sum of an infinite geometric series can be found with a formula that uses S for the sum, a for the first term, and r for the common ratio.

This method gives a fast way to calculate long-term results in processes with exponential decrease. It is used in physics, economics, and engineering, where decay models help with prediction and analysis.

Infinite Geometric Series and Decay

sequences

Recursive Sequences and Partial Sums

Sequences are ordered lists of numbers that follow a rule or pattern. Each number in the list is called a term. Terms are often written as a1, a2, a3, and so on, where the subscript shows the position in the sequence. When the pattern is clear, dots may be used to show that the list continues. A sequence can also be viewed as a function. Its domain is the set of natural numbers, and each natural number maps to one term in the list. This view is useful when a sequence is defined explicitly or...

summation notation

Summation Notation for Sequences

Sigma notation, also called summation notation, gives a short way to write the sum of a sequence of terms that follow a regular pattern. It uses the uppercase Greek letter sigma, ∑, to show that terms are being added together. A typical sigma expression includes an index of summation. The index starts at a lower value, such as 1, and ends at an upper value, such as n. The term a_k represents the general term in the sequence, where k changes across the stated range. For example, the increasing...

arithmetic sequences

Arithmetic Sequence Formulas and Partial Sums

Arithmetic sequences follow a regular number pattern. Each term is found by adding the same value, called the common difference, to the term before it. This steady change makes the sequence easy to study and predict. The nth-term formula helps find any term in the sequence without listing every earlier number. In this formula, a n is the nth term, a is the first term, d is the common difference, and n shows the term position. It gives a direct way to calculate a specific value from the pattern.

geometric sequences

Infinite geometric series describe values that shrink by the same factor each step. Each new term is found by multiplying the previous term by a fixed common ratio. This pattern is useful for modeling gradual loss, such as energy dissipation or a drop in amplitude over time. When the common ratio has a magnitude less than one, the terms keep getting smaller. In that case, the total of the series approaches a definite number instead of growing without bound or changing back and forth. This is...

mathematical induction

Proving Number Patterns with Induction

Mathematical induction is a proof method for statements about natural numbers. It is often used to test whether a pattern really works for every number in a sequence. A common example is the sum of the first n natural numbers, where the formula seems to hold as more terms are added. The proof begins with a base case. In this step, the formula is checked at the first value, usually n = 1. Substituting 1 into both sides shows that the equality is true, so the base case holds. Next comes the...

the binomial theorem

Expanding Binomials with the Binomial Theorem

The Binomial Theorem gives a clear method for expanding binomial expressions raised to a power. It applies to binomials of the form (a + b)^n, where a and b are variables or constants that represent algebraic expressions, and n is a non-negative integer. Each term in the expansion follows a fixed pattern. The binomial coefficient in each term is found using factorials. The exponent of a decreases from n to 0, while the exponent of b increases from 0 to n. The theorem also helps students...

binomial expansion using pascal's triangle

binomial-expansion-using-pascal-s-triangle

Pascal’s Triangle for Binomial Coefficients

Pascal’s Triangle helps organize the coefficients in a binomial expansion. A binomial is an expression with two terms, such as (a + b), and raising it to a power creates a predictable pattern of terms. The triangle is built row by row. The top row, called the zeroth row, matches (a + b)0. Each new row gives the coefficients for the next power of the binomial, so the sixth row, 1, 5, 10, 10, 5, 1, gives the coefficients for (a + b)5. Each row begins and ends with 1. Every number inside the...

key terms and concepts

Learn how infinite geometric series model repeated decay, when they converge, and how the sum formula helps solve real-world problems in science and math.

Infinite Geometric Series and Decay | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Infinite Geometric Series and Decay

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