infinite

Examples of infinite in the following topics:

• Summing an Infinite Series

  • Infinite sequences and series can either converge or diverge.
  • If the limit of is infinite or does not exist, the series is said to diverge.
  • An easy way that an infinite series can converge is if all the $a_{n}$ are zero for sufficiently large $n$s.
  • Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.
  • An infinite sequence of real numbers shown in blue dots.

• 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:

• Cylindrical Shells

  • In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.
  • Intuitively speaking, part of the graph of a function is rotated around an axis, and is modeled by an infinite number of cylindrical shells, all infinitely thin.
  • By adding the volumes of all these infinitely thin cylinders, we can calculate the volume of the solid formed by the revolution.

• Series

  • Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
  • Given an infinite sequence of numbers $\{ a_n \}$, a series is informally the result of adding all those terms together: $a_1 + a_2 + a_3 + \cdots$ .
  • As there are an infinite number of terms, this notion is often called an infinite series.
  • Unlike finite summations, infinite series need tools from mathematical analysis, specifically the notion of limits, to be fully understood and manipulated.
  • In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, and finance.

• Expressing Functions as Power Functions

  • Since all infinitely differentiable functions can be represented in power series, any infinitely differentiable function can be represented as a sum of many power functions (of integer exponents).
  • The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number $a$ is the power series:
  • Therefore, an arbitrary function that is infinitely differentiable is expressed as an infinite sum of power functions ($x^n$) of integer exponent.
  • Describe the relationship between the power functions and infinitely differentiable functions

• The Integral Test and Estimates of Sums

  • The integral test is a method of testing infinite series of nonnegative terms for convergence by comparing them to an improper integral.
  • The integral test for convergence is a method used to test infinite series of non-negative terms for convergence.
  • The infinite series $\sum_{n=N}^\infty f(n)$ converges to a real number if and only if the improper integral $\int_N^\infty f(x)\,dx$ is finite.
  • In this way, it is possible to investigate the borderline between divergence and convergence of infinite series.
  • Since the area under the curve $y = \frac{1}{x}$ for $x \in [1, \infty)$ is infinite, the total area of the rectangles must be infinite as well.

• Spherical and Plane Waves

  • Spherical waves come from point source in a spherical pattern; plane waves are infinite parallel planes normal to the phase velocity vector.
  • A plane wave is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector .
  • It is not possible in practice to have a true plane wave; only a plane wave of infinite extent will propagate as a plane wave.
  • Plane waves are an infinite number of wavefronts normal to the direction of the propogation.

• Taylor Polynomials

  • A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function's derivatives.
  • Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
  • The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number $a$ is the power series
  • Therefore, as long as Taylor expansion is possible and the infinite sum converges, the definite integral ($I$) can be evaluated.

• Tips for Testing Series

  • Convergence tests are methods of testing for the convergence or divergence of an infinite series.
  • Convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series.
  • Since the area under the curve $y = \frac{1}{x}$ for $x \in [1, ∞)$ is infinite, the total area of the rectangles must be infinite as well.

• Comparison Tests

  • The limit comparison test is a method of testing for the convergence of an infinite series, while the direct comparison test is a way of deducing the convergence or divergence of an infinite series or an improper integral by comparison with other series or integral whose convergence properties are already known.
  • The direct comparison test provides a way of deducing the convergence or divergence of an infinite series or an improper integral.
  • If the infinite series $\sum b_n$ converges and $0 \le a_n \le b_n$ for all sufficiently large $n$ (that is, for all $n>N$ for some fixed value $N$), then the infinite series $\sum a_n$ also converges.
  • If the infinite series $\sum b_n$ diverges and $a_n \ge b_n \ge 0$ for all sufficiently large $n$, then the infinite series $\sum a_n$ also diverges.