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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.
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:
This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition.
Once such a sequence is found, a similar question can be asked with f(n) taking the role of 1/n, and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series. Using the integral test for convergence, one can show (see below) that, for every natural number k, the series
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
The proof is the same as for complex-valued series: use the completeness to derive the Cauchy criterion for convergence—a series is convergent if and only if its tails can be made arbitrarily small in norm—and apply the triangle inequality. In particular, for series with values in any Banach space, absolute convergence implies convergence ...
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.It depends on the quantity | |, where are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one.