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The sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the sum: + + + + = This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2: (+)
It may be used to prove Nicomachus's theorem that the sum of the first cubes equals the square of the sum of the first positive integers. [2] Summation by parts is frequently used to prove Abel's theorem and Dirichlet's test.
The general form of the theorem was first conjectured by Legendre in his attempted unsuccessful proofs of quadratic reciprocity [5] — as Gauss noted in his Disquisitiones Arithmeticae [6] — but it was proved by Dirichlet with Dirichlet L-series. The proof is modeled on Euler's earlier work relating the Riemann zeta function to the ...
Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when x is an integer. The integral is not a convergent Lebesgue integral ; it is understood as the Cauchy principal value .
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or equivalently the Dirichlet convolution of an arithmetic function () with one:
An arithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various applications, such as the computation of expected values in probability theory , especially in Bernoulli processes .
We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c.. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0)Each equation follows by definition [A1]; the first with a + b, the second with b.
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 π 2 / 6 {\displaystyle \pi ^{2}/6} and announced this discovery in 1735.
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