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The depleted harmonic series where all of the terms in which the digit 9 appears anywhere in the denominator are removed can be shown to converge to the value 22.92067 66192 64150 34816.... [44] In fact, when all the terms containing any particular string of digits (in any base) are removed, the series converges. [45]
Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a n converge to 0 monotonically, but this condition is not necessary for convergence.
This was proved by Leonhard Euler in 1737, [1] and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series).
A classic example is the alternating harmonic series given by + + = = +, which converges to (), but is not absolutely convergent (see Harmonic series). Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see Riemann series theorem .
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
A particular case of Dirichlet's test is the more commonly used alternating series test for the case [2] [5] = | = | Another corollary is that ∑ n = 1 ∞ a n sin n {\textstyle \sum _{n=1}^{\infty }a_{n}\sin n} converges whenever ( a n ) {\displaystyle (a_{n})} is a decreasing sequence that tends to zero.
where either all a n are positive or all a n are negative, is called an alternating series. The alternating series test guarantees that an alternating series converges if the following two conditions are met: [1] [2] [3] | | decreases monotonically [a], i.e., | + | | |, and
2 Proof. 3 Example. 4 One-sided version. ... Alternating series; Cauchy condensation; Dirichlet; Abel; ... we have that the original series also converges. One-sided ...