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In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original ...
A tabulation of the numbers in a triangular array is called the Euler triangle or Euler's triangle. It shares some common characteristics with Pascal's triangle . Values of A ( n , k ) {\textstyle A(n,k)} (sequence A008292 in the OEIS ) for 0 ≤ n ≤ 9 {\textstyle 0\leq n\leq 9} are:
where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory. [3]
The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ. The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms.
The sum of the series is approximately equal to 1.644934. [3] The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be / and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he ...
This book describes recursive models applied to theoretical questions in monetary policy, fiscal policy, taxation, economic growth, search theory, and labor economics. In investment and finance, Avinash Dixit and Robert Pindyck showed the value of the method for thinking about capital budgeting , in particular showing how it was theoretically ...
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler .
In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n many k th powers of positive integers is itself a k th power, then n is greater than or equal to k :