geometric series

Examples of geometric series in the following topics:

• Infinite Geometric Series

  • Geometric series are one of the simplest examples of infinite series with finite sums.
  • A geometric series is an infinite series whose terms are in a geometric progression, or whose successive terms have a common ratio.
  • If the terms of a geometric series approach zero, the sum of its terms will be finite.
  • A geometric series with a finite sum is said to converge.
  • Find the sum of the infinite geometric series $64+ 32 + 16 + 8 + \cdots$

• Applications of Geometric Series

  • Geometric series have applications in math and science and are one of the simplest examples of infinite series with finite sums.
  • Geometric series are used throughout mathematics.
  • Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have this property.
  • In the case of the Koch snowflake, its area can be described with a geometric series.
  • Apply geometric sequences and series to different physical and mathematical topics

• Sums and Series

  • If you add up all the terms of a geometric sequence (one in which each entry is the previous entry multiplied by a constant), you have a geometric series.
  • The common example of a trend that follows a geometric series is the number of people infected with a virus, as each person passes it to several more.
  • Once again, pause to convince yourself that this will work on all geometric series, but only on geometric series.
  • Finally—once again—we can apply this trick to the generic geometric series to find a formula.
  • So the total number of people infected follows a geometric series.

• Summing the First n Terms in a Geometric Sequence

  • Geometric series are examples of infinite series with finite sums, although not all of them have this property.
  • Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of the convergence of series.
  • The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant.
  • The following are several geometric series with different common ratios.
  • For $r\neq 1$, the sum of the first $n$ terms of a geometric series is:

• Geometric Sequences

  • A geometric progression, also known as a geometric sequence, is an ordered list of numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio $r$.
  • The $n$th term of a geometric sequence with initial value $a$ and common ratio $r$ is given by
  • The common ratio of a geometric series may be negative, resulting in an alternating sequence.
  • For instance: $1,-3,9,-27,81,-243, \cdots$ is a geometric sequence with common ratio $-3$.
  • The behavior of a geometric sequence depends on the value of the common ratio.

• Taylor and Maclaurin Series

  • If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
  • A function may not be equal to its Taylor series, even if its Taylor series converges at every point.
  • In the case that $a=0$, the series is also called a Maclaurin series.
  • The Maclaurin series for $(1 − x)^{−1}$ for $\left| x \right| < 1$ is the geometric series: $1+x+x^2+x^3+\cdots\!
  • Identify a Maclaurin series as a special case of a Taylor series

• Recursive Definitions

  • An applied example of a geometric sequence involves the spread of the flu virus.
  • Suppose each infected person will infect two more people, such that the terms follow a geometric sequence.
  • Using this equation, the recursive equation for this geometric sequence is:
  • One can work out every term in the series just by knowing previous terms.
  • Each person infects two more people with the flu virus, making the number of recently-infected people the nth term in a geometric sequence.

• Sculpture of the Cyclades

  • Cycladic art during the Greek Bronze Age is noted for its abstract, geometric designs of male and female figures.
  • From the late fourth millennium BCE to the early second millennium BCE, Cycladic sculptures went through a series of stylistic shifts, with bodily forms varying from geometric to organic.
  • These figures are based in simple geometric shapes.
  • Like the female figures, the shape of the male figure is reliant on geometric shapes and flat planes.

• Convergence of Series with Positive Terms

  • The series $\sum_{n \ge 1} \frac{1}{n^2}$ is convergent because of the inequality:
  • In other words, the series has an upper bound.
  • Proving that the series is equal to $2$ requires only elementary algebra, however.
  • If the series is denoted $S$, it can be seen that:
  • Visualization of the geometric sum in Example 2.

• Introduction to geometric distribution (special topic)

  • These questions can be answered using the geometric distribution.
  • We first formalize each trial – such as a single coin flip or die toss – using the Bernoulli distribution, and then we combine these with our tools from probability (Chapter 2) to construct the geometric distribution.