More recently, Wang and Wen [4] strengthened Theorem 1 to the finite sum case. The sum is called the non-alternating summation of order . In both the cases 0< 1< 2<⋯< −1< . A … The second sum is called the alternating sum of order . Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. $\begingroup$ A cultural note: Euler was (as far as I know) the first person to observe that this series diverges (and without assuming a priori that it was an infinite series), thus obtaining a new proof of the infinitude of the primes. ORDER 2 We consider first the problem of finding the following second order sums. Alternating Sums of the Reciprocals of Binomial Coefficients Hac`ene Belbachir and Mourad Rahmani University of Sciences and Technology Houari Boumediene Faculty of Mathematics P. O. The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. This sum definitely converges by comparison to $1/n^2$ but I was wondering if it was an important constant and/or the value of a specific notable function. Equation 1: Sum of the reciprocals of even powers of integer numbers. called “twin primes” if both p and p+2 are prime numbers. Alternating Sums of the Reciprocals of Binomial Coe cients Hac ene Belbachir and Mourad Rahmani University of Sciences and Technology Houari Boumediene Faculty of Mathematics Po. Then the p i will also include all the consecutive primes after p n, up until a point where their reciprocal sum is greater than s (since the sum of the reciprocals of the primes diverges) but less than r (since 1/p j for j > n is always less than 1/p n, the first prime that puts the sum over s will not put it … Finally the sum of the entries in T is the acceptable value for the alternating sum of the primes under "Eulerian summation", if this sum converges. it gives more information than) Euclid's 3rd-century-BC result that there are infinitely many prime numbers.. ( , )=∑ 1 + + =1 ( , )=∑ (−1) There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that This result extends the Basel problem from exponent 2 to any even exponent. The sum of the reciprocals of all prime numbers diverges; that is: ∑ = + + + + + + + ⋯ = ∞ This was proved by Leonhard Euler in 1737, and strengthens (i.e. $$ \sum_{p\in \mathbb{P}} \frac{1}{p^2} $$ where $\mathbb{P}$ is the set of primes. The twin prime problem is the famous question of whether there are an infinite number of twin primes. Although this problem is unsolved, in 1919 Viggo Brun proved the remarkable theorem that the sum of the reciprocals of the twin primes converges. One knows that the sum of the reciprocal of all odd integers diverges and so it is reasonable to also conclude that the sum of the reciprocal of the first powers of all primes also diverges. Lets next look at the product of the first x primes. Euler’s astonishingly clever method “has fascinated mathematicians ever since.”Euler had previously proved the Basel problem in 1734. In this article, we focus on the alternating sums of the reciprocal Fibonacci numbers Xmn k=n (−1)k F ak+b, where a∈ {1,2,3} and b

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