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This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with ellipsis leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of ...
To emphasize that there are an infinite number of terms, series are often also called infinite series. Series are represented by an expression like a 1 + a 2 + a 3 + ⋯ , {\displaystyle a_{1}+a_{2}+a_{3}+\cdots ,} or, using capital-sigma summation notation , [ 8 ] ∑ i = 1 ∞ a i . {\displaystyle \sum _{i=1}^{\infty }a_{i}.}
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.
This last non-simple continued fraction (sequence A110185 in the OEIS), equivalent to = [;,,,,,...], has a quicker convergence rate compared to Euler's continued fraction formula [clarification needed] and is a special case of a general formula for the exponential function:
The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. ... The infinite series ...
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
Yet in the limit the sequence {Τ n} defines an infinite continued fraction which (if it converges) represents a single point in the complex plane. When an infinite continued fraction converges, the corresponding sequence {Τ n} of LFTs "focuses" the plane in the direction of x, the value of the continued fraction.
Moreover, in the standard decimal representation of , an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself if is an integer. Certain procedures for constructing the decimal expansion of x {\displaystyle x} will avoid the problem of trailing 9's.