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While the partial sums of the reciprocals of the primes eventually exceed any integer value, they never equal an integer. One proof [6] is by induction: The first partial sum is 1 / 2 , which has the form odd / even . If the n th partial sum (for n ≥ 1) has the form odd / even , then the (n + 1) st sum is
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. [1]
Rules for calculating the periods of repeating decimals from rational fractions were given by James Whitbread Lee Glaisher in 1878. [5] For a prime p, the period of its reciprocal divides p − 1. [6] The sequence of recurrence periods of the reciprocal primes (sequence A002371 in the OEIS) appears in the 1973 Handbook of Integer Sequences.
It is a divergent series: as more terms of the series are included in partial sums of the series, the values of these partial sums grow arbitrarily large, beyond any finite limit. Because it is a divergent series, it should be interpreted as a formal sum, an abstract mathematical expression combining the unit fractions, rather than as something ...
The two classical summation methods for series, ordinary convergence and absolute convergence, define the sum as a limit of certain partial sums. These are included only for completeness; strictly speaking they are not true summation methods for divergent series since, by definition, a series is divergent only if these methods do not work.
Another way to combine two (disjoint) posets is the ordinal sum [12] (or linear sum), [13] Z = X ⊕ Y, defined on the union of the underlying sets X and Y by the order a ≤ Z b if and only if: a, b ∈ X with a ≤ X b, or; a, b ∈ Y with a ≤ Y b, or; a ∈ X and b ∈ Y. If two posets are well-ordered, then so is their ordinal sum. [14]
Using the symbols , for the partial sums of the original series and , for the partial sums of the series after multiplication by , this definition implies that , =, for all , and therefore also , =,, when the limits exist. Therefore if a series is summable, any nonzero scalar multiple of the series is also summable and vice versa: if a series ...
This page was last edited on 19 December 2004, at 20:24 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.
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