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The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2]
[8] [9] [10] Bernoulli credited his brother Johann Bernoulli for finding the proof, [10] and it was later included in Johann Bernoulli's collected works. [11] The partial sums of the harmonic series were named harmonic numbers, and given their usual notation , in 1968 by Donald Knuth. [12]
The infinite series whose terms are the natural numbers 1 + 2 + 3 ... fails to converge to a finite limit, ... 10, 15, etc. The nth partial sum is given by a simple ...
When the limit of the sequence exists, the real number L is the limit of this sequence if and only if for every real number ε > 0, there exists a natural number N such that for all n > N, we have | a n − L | < ε. [9] The common notation = is read as: "The limit of a n as n approaches infinity equals L" or "The limit as n approaches infinity ...
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
In mathematical analysis, Cesàro summation (also known as the Cesàro mean [1] [2] or Cesàro limit [3]) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.
if L > 1 the series converges (this includes the case L = ∞) if L < 1 the series diverges; and if L = 1 the test is inconclusive. An alternative formulation of this test is as follows. Let { a n} be a series of real numbers. Then if b > 1 and K (a natural number) exist such that
Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the Riemann series theorem. [31] [32] [33]