finite

Examples of finite in the following topics:

• The Effect of the Finite Volume

  • Real gases deviate from the ideal gas law due to the finite volume occupied by individual gas particles.
  • The particles of a real gas do, in fact, occupy a finite, measurable volume.
  • At high pressures, the deviation from ideal behavior occurs because the finite volume that the gas molecules occupy is significant compared to the total volume of the container.

• Adiabatic Processes

  • It is impossible to reduce the temperature of any system to zero temperature in a finite number of finite operations.
  • Assuming an entropy difference at absolute zero, T=0 could be reached in a finite number of steps.
  • Left side: Absolute zero can be reached in a finite number of steps if S(T=0,X1)≠S(T=0, X2).

• Summing Terms in an Arithmetic Sequence

  • An arithmetic sequence which is finite has a specific formula for its sum.
  • The sum of the members of a finite arithmetic sequence is called an arithmetic series.
  • We can come up with a formula for the sum of a finite arithmetic formula by looking at the sum in two different ways.
  • 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.

• Infinite Geometric Series

  • Geometric series are one of the simplest examples of infinite series with finite sums.
  • 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.
  • What follows in an example of an infinite series with a finite sum.
  • If a series converges, we want to find the sum of not only a finite number of terms, but all of them.

• Summing an Infinite Series

  • Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
  • Unlike finite summations, infinite series need tools from mathematical analysis, and specifically the notion of limits, to be fully understood and manipulated.
  • A series is said to converge when the sequence of partial sums has a finite limit.
  • Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

• Introduction to Sequences

  • 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.
  • A more formal definition of a finite sequence with terms in a set $S$ is a function from $\left \{ 1, 2, \cdots, n \right \}$ to $S$ for some $n > 0$.
  • A sequence of a finite length n is also called an $n$-tuple.
  • Finite sequences include the empty sequence $( \quad )$ that has no elements.

• Absolute Convergence and Ratio and Root Tests

  • An infinite series of numbers is said to converge absolutely if the sum of the absolute value of the summand is finite.
  • An infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute value of the summand is finite.
  • Similarly, an improper integral of a function, $\textstyle\int_0^\infty f(x)\,dx$, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if $\int_0^\infty \left|f(x)\right|dx = L$.
  • Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess, yet is broad enough to occur commonly.

• Counting Rules and Techniques

  • Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
  • Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
  • A bijective proof is a proof technique that finds a bijective function $f: A \rightarrow B$ between two finite sets $A$ and $B$, which proves that they have the same number of elements, $|A| = |B|$.
  • In this technique, a finite set $X$ is described from two perspectives, leading to two distinct expressions for the size of the set.

• The Wave Function

  • This is because the values of the wave function and its first order derivatives may not be finite and definite (having exactly one value), which means that the probabilities can be infinite and have multiple values at any one position and time, which is nonsense.
  • Furthermore, when we use the wave function to calculate an observation of the quantum system without meeting these requirements, there will not be finite or definite values to use (in this case the observation can take a number of values and can be infinite).

• The Importance of Market Segmentation

  • Segmentation splits buyers into groups with similar needs and wants to best utilize a firm's finite resources through buyer based marketing.
  • Market segmentation allows for a better allocation of a firm's finite resources.