Search results
Results from the WOW.Com Content Network
In mathematics, a telescoping series is a series whose general term is of the form = +, i.e. the difference of two consecutive terms of a sequence (). As a consequence the partial sums of the series only consists of two terms of ( a n ) {\displaystyle (a_{n})} after cancellation.
The geometric series on the real line. In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits, it may be possible to assign a value to a series, called the sum of the series.
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.
The sum of the reciprocals of all the Fermat numbers (numbers of the form + ) (sequence A051158 in the OEIS) is irrational. The sum of the reciprocals of the pronic numbers (products of two consecutive integers) (excluding 0) is 1 (see Telescoping series).
The partial sum of the first n positive pronic numbers is twice the value of the n th tetrahedral number: = (+) = (+) (+) =. The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1: [7]
The sum of the series is a random variable whose probability density function is close to for values between and , and decreases to near-zero for values greater than or less than . Intermediate between these ranges, at the values ± 2 {\displaystyle \pm 2} , the probability density is 1 8 − ε {\displaystyle {\tfrac {1}{8}}-\varepsilon } for ...
Te n is the sum of all products p × q where (p, q) are ordered pairs and p + q = n + 1 Te n is the number of ( n + 2)-bit numbers that contain two runs of 1's in their binary expansion. The largest tetrahedral number of the form 2 a + 3 b + 1 {\displaystyle 2^{a}+3^{b}+1} for some integers a {\displaystyle a} and b {\displaystyle b} is 8436 .