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.

• 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.

• Applications and Problem-Solving

  • Arithmetic series can simplify otherwise complex addition problems by decreasing the number of terms to be added.
  • For example, let's say we wanted to write the series of all the even numbers between 50 and 100.
  • This trick applies to all arithmetic series.
  • As long as you go up by the same amount as you go down, the sum will stay the same—and this is just what happens for arithmetic series.

• Series and Sigma Notation

  • For example, $4+9+3+2+17$ is a series.
  • This particular series adds up to $35$.
  • Another series is $2+4+8+16+32+64$.
  • This series sums to $126$.
  • It indicates a series.

• Complex Numbers in Polar Coordinates

  • This leads to a way to visualize multiplying and dividing complex numbers geometrically.
  • Sometimes it is helpful to think of complex numbers in a different geometric way.
  • The previous geometric idea where the number $z=a+bi$ is associated with the point $(a,b)$ on the usual $xy$-coordinate system is called rectangular coordinates.
  • This way of thinking about multiplying and dividing complex numbers gives a geometric way of thinking about those operations.

• Complex Conjugates

  • Two complex conjugates of each other multiply to be a real number with geometric significance.