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Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs and how to get involved. Comments: 12 pages, 9 figures, 1 table Subjects: Combinatorics math.

This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. See Faulhaber's formula. See zeta constants. The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form :. Sums of sines and cosines arise in Fourier series. From Wikipedia, the free encyclopedia.

Wikipedia list article. Wolfram Research, Inc. Archived from the original on Retrieved Academic Press, Inc. Wolfram Research. Retrieved 2 June Categories : Mathematical series Mathematics-related lists Mathematical tables. Hidden categories: Articles with short description Short description is different from Wikidata.

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The n th partial sum of the series is the triangular number. Because the sequence of partial sums fails to converge to a finite limit , the series does not have a sum. Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. These methods have applications in other fields such as complex analysis , quantum field theory , and string theory. In a monograph on moonshine theory , Terry Gannon calls this equation "one of the most remarkable formulae in science".

Home About Us Services Contact. Computation: Finite and Infinite Machines. Though only some of us deal directly with computers, all of us are falling under the shadow of their ever-growing sphere of influence, and thus we all need to understand their capabilities and their limitations. It may take up to minutes before you receive it. In order to navigate out of this carousel please use your heading shortcut key to navigate to the next or previous heading. Computation: Finite and Infinite Machines Marvin Lee Minsky Man has within a single generation found himself sharing the world with a strange new species: the computers and computer-like machines. Semantic Information Processing Please read our short guide Access codes and supplements are not guaranteed with used items.

This is a finite summation and offers no difficulties. If the partial sums si converge to a finite limit as i → ∞, lim i→∞ si = S,. () eq the infinite series. ∑∞.

For instance, a function call can terminate and return a value, as well as have output effects during its execution. Here, we deal with semantic definitions covering both results and observations. Often, such definitions are provided for finite computations only.

We now consider what happens when we add an infinite number of terms together. Surely if we sum infinitely many numbers, no matter how small they are, the answer goes to infinity? In some cases the answer does indeed go to infinity like when we sum all the positive integers , but surprisingly there are some cases where the answer is a finite real number. If the sum of a series gets closer and closer to a certain value as we increase the number of terms in the sum, we say that the series converges. In other words, there is a limit to the sum of a converging series.

*Convergence , in mathematics , property exhibited by certain infinite series and functions of approaching a limit more and more closely as an argument variable of the function increases or decreases or as the number of terms of the series increases. Although no finite value of x will cause the value of y to actually become zero, the limiting value of y is zero because y can be made as small as desired by choosing x large enough.*

We also show a proof using Algebra below. We often use Sigma Notation for infinite series. Our example from above looks like:. Let's add the terms one at a time. When the "sum so far" approaches a finite value, the series is said to be " convergent ":. When the difference between each term and the next is a constant, it is called an arithmetic series. When the ratio between each term and the next is a constant, it is called a geometric series.

In mathematics , the harmonic series is the divergent infinite series. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music. The divergence of the harmonic series was first proven in the 14th century by Nicole Oresme , [1] but this achievement fell into obscurity.

While the English words "sequence" and "series" have similar meanings, in mathematics they are completely different concepts. A sequence is a list of numbers placed in a defined order while a series is the sum of such a list of numbers. There are many kinds of sequences, including those based on infinite lists of numbers. Different sequences and the corresponding series have different properties and can give surprising results. Sequences are lists of numbers placed in a definite order according to given rules. The series corresponding to a sequence is the sum of the numbers in that sequence.

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## 3 Comments

## Kelly B.

Sequences and Series; Finite and Infinite. Calculus 12, Veritas Prep. We are about to do the coolest theorem in calculus. Not the most important theorem, mind.

## Ella F.

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## Amber P.

of sequential limit, namely the limit (or sum) of a series. Zeno's paradox. How can an infinite number of things happen in a finite amount of time?