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A series is, informally speaking, the sum of the terms of a sequence. That is, it is an expression of the form ∑ n = 1 ∞ a n {\textstyle \sum _{n=1}^{\infty }a_{n}} or a 1 + a 2 + ⋯ {\displaystyle a_{1}+a_{2}+\cdots } , where ( a n ) {\displaystyle (a_{n})} is a sequence of real or complex numbers.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
"A comma goes before 'and' or 'or' in a series of three or more: Sn, K, Na, and Li lines are invisible." Plain English Handbook, Revised Edition (McCormick-Mathers Publishing Co., 1959), § 483, p. 78 "Use commas to separate the items in a series of words, phrases, or short clauses: The farmer sold corn, hay, oats, potatoes, and wheat."
However, if the original series diverges, then the grouped series do not necessarily diverge, as in this example of Grandi's series above. However, divergence of a grouped series does imply the original series must be divergent, since it proves there is a subsequence of the partial sums of the original series which is not convergent, which ...
In linguistics and grammar, a sentence is a linguistic expression, such as the English example "The quick brown fox jumps over the lazy dog." In traditional grammar , it is typically defined as a string of words that expresses a complete thought, or as a unit consisting of a subject and predicate .
The ellipsis (/ ə ˈ l ɪ p s ɪ s /, plural ellipses; from Ancient Greek: ἔλλειψις, élleipsis, lit. ' leave out ' [1]), rendered ..., alternatively described as suspension points [2]: 19 /dots, points [2]: 19 /periods of ellipsis, or ellipsis points, [2]: 19 or colloquially, dot-dot-dot, [3] [4] is a punctuation mark consisting of a series of three dots.
In this example, predictions for the weather on more distant days change less and less on each subsequent day and tend towards a steady state vector. [5] This vector represents the probabilities of sunny and rainy weather on all days, and is independent of the initial weather. [5] The steady state vector is defined as:
The first example contains the long verb phrase hit the ball well enough to win their first World Series since 2000; the second is a verb phrase composed of the main verb saw, the complement phrase the man (a noun phrase), and the adjunct phrase through the window (an adverbial phrase and prepositional phrase).