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A generalized version of the conjecture states that, for any positive , all but finitely many fractions can be expressed as a sum of three positive unit fractions. The conjecture for fractions 5 n {\displaystyle {\tfrac {5}{n}}} was made by Wacław Sierpiński in a 1956 paper, which went on to credit the full conjecture to Sierpiński's student ...
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]
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
In chemistry, the mass fraction of a substance within a mixture is the ratio (alternatively denoted ) of the mass of that substance to the total mass of the mixture. [1] Expressed as a formula, the mass fraction is: =. Because the individual masses of the ingredients of a mixture sum to , their mass fractions sum to unity: = =
The simplest fraction 3 / y with a three-term expansion is 3 / 7 . A fraction 4 / y requires four terms in its greedy expansion if and only if y ≡ 1 or 17 (mod 24), for then the numerator −y mod x of the remaining fraction is 3 and the denominator is 1 (mod 6). The simplest fraction 4 / y with a four-term ...
The sum of all volume fractions of a mixture is equal to 1: ∑ i = 1 N V i = V ; ∑ i = 1 N φ i = 1. {\displaystyle \sum _{i=1}^{N}V_{i}=V;\qquad \sum _{i=1}^{N}\varphi _{i}=1.} The volume fraction (percentage by volume, vol%) is one way of expressing the composition of a mixture with a dimensionless quantity ; mass fraction (percentage by ...
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. . Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or i
Mole fraction is numerically identical to the number fraction, which is defined as the number of particles of a constituent N i divided by the total number of all molecules N tot. Whereas mole fraction is a ratio of amounts to amounts (in units of moles per moles), molar concentration is a quotient of amount to volume (in units of moles per litre).
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