infinite

Examples of infinite in the following topics:

• Infinite Geometric Series

  • Geometric series are one of the simplest examples of infinite series with finite sums.
  • What follows in an example of an infinite series with a finite sum.
  • Applying $r^n\rightarrow 0$, we can find a new formula for the sum of an infinitely long geometric series:
  • Find the sum of the infinite geometric series $64+ 32 + 16 + 8 + \cdots$
  • Substitute $a=64$ and $\displaystyle r= \frac{1}{2}$ into the formula for the sum of an infinite geometric series:

• Summing Terms in an Arithmetic Sequence

  • An infinite arithmetic series is exactly what it sounds like: an infinite series whose terms are in an arithmetic sequence.
  • The general form for an infinite arithmetic series is:
  • If either $a_1$ or $d$ is non-zero, then the infinite series has no sum.
  • Even if one is dealing with an infinite sequence, the sum of that sequence can still be found up to any $n$th term with the same equation used in a finite arithmetic sequence.

• Sequences of Mathematical Statements

  • The length of a sequence is the number of ordered elements, and it may be infinite.
  • Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2,4,6, \cdots )$.
  • It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.

• Applications of Geometric Series

  • His method was to dissect the area into an infinite number of triangles.
  • Assuming that the blue triangle has area $1$, the total area is an infinite series:
  • The Koch snowflake is a fractal shape with an interior comprised of an infinite amount of triangles.
  • Zeno's mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite.
  • The interior of a Koch snowflake is comprised of an infinite amount of triangles.

• Inconsistent and Dependent Systems in Three Variables

  • Dependent systems have an infinite number of solutions.
  • We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions.
  • An infinite number of solutions can result from several situations.
  • The result we get is an identity, $0 = 0$, which tells us that this system has an infinite number of solutions.
  • The solution set is infinite, as all points along the intersection line will satisfy all three equations.

• Inconsistent and Dependent Systems in Two Variables

  • For linear equations in two variables, inconsistent systems have no solution, while dependent systems have infinitely many solutions.
  • An inconsistent system has no solution, and a dependent system has an infinite number of solutions.
  • Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line.
  • Note that there are an infinite number of solutions to a dependent system, and these solutions fall on the shared line.

• Introduction to Sequences

  • The number of ordered elements (possibly infinite) is called the length of the sequence.
  • Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2, 4, 6, \cdots )$.
  • Finite sequences are sometimes known as strings or words and infinite sequences as streams.
  • An infinite sequence in $S$ is a function from $\left \{ 1, 2, \cdots \right \}$ to $S$.
  • Part of an infinite sequence of real numbers (in blue).

• Proof by Mathematical Induction

  • Proving an infinite sequence of statements is necessary for proof by induction, a rigorous form of deductive reasoning.
  • It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.

• Solving Problems with Inequalities

  • Note that the expression x > 12 has infinitely many solutions.
  • A linear equation, we know, may have exactly one solution, infinitely many solutions, or no solution.
  • A linear inequality may have infinitely many solutions or no solutions.

• Introduction to Systems of Equations

  • Some linear systems may not have a solution, while others may have an infinite number of solutions.
  • A dependent system has infinitely many solutions.