Examples of infinitive in the following topics:
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- 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 an are zero for sufficiently large ns.
- 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.
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- 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 rn→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+⋯
- Substitute a=64 and r=21 into the formula for the sum of an infinite geometric series:
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- 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.
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- Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
- Given an infinite sequence of numbers {an}, a series is informally the result of adding all those terms together: a1+a2+a3+⋯ .
- 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.
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- 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 (xn) of integer exponent.
- Describe the relationship between the power functions and infinitely differentiable functions
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- 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 ∑n=N∞f(n) converges to a real number if and only if the improper integral ∫N∞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=x1 for x∈[1,∞) is infinite, the total area of the rectangles must be infinite as well.
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- 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.
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- 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.
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- 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=x1 for $x \in [1, ∞)$ is infinite, the total area of the rectangles must be infinite as well.
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- 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 ∑bn converges and 0≤an≤bn for all sufficiently large n (that is, for all n>N for some fixed value N), then the infinite series ∑an also converges.
- If the infinite series ∑bn diverges and an≥bn≥0 for all sufficiently large n, then the infinite series ∑an also diverges.