Convergent series information

{\displaystyle S_{n}=a_{1}+a_{2}+\cdots +a_{n}=\sum _{k=1}^{n}a_{k}.} A series is convergent (or converges) if and only if the sequence ( S 1 , S 2 , S 3 , …...

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the...

refer to: Convergent boundary, a type of plate tectonic boundary Convergent (continued fraction) Convergent evolution Convergent series Convergent may also...

D_{R}} , the series is also uniformly convergent on S . {\displaystyle S.} Every uniformly convergent sequence is locally uniformly convergent. Every locally...

mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely. More precisely, a series of real...

creates a series with the same convergence as the original series. Absolutely convergent series are unconditionally convergent. But the Riemann series theorem...

similar, equivalent series was published by Joseph Ser in 1926. In 1997 K. Maślanka gave another globally convergent (except s = 1) series for the Riemann...

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...

} is not a power series. A power series ∑ n = 0 ∞ a n ( x − c ) n {\textstyle \sum _{n=0}^{\infty }a_{n}(x-c)^{n}} is convergent for some values of...

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...

A convergent boundary (also known as a destructive boundary) is an area on Earth where two or more lithospheric plates collide. One plate eventually slides...

a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have...

{x^{4}}{4!\cdot 4}}\mp \cdots } These series are convergent at any complex x, although for |x| ≫ 1, the series will converge slowly initially, requiring...

5} , the result is inaccurate due to cancellation. A faster converging series was found by Ramanujan: E i ( x ) = γ + ln ⁡ x + exp ⁡ ( x / 2 ) ∑ n = 1...

Convergent evolution is the independent evolution of similar features in species of different periods or epochs in time. Convergent evolution creates...

summation of convergent series, but it has interesting properties, such as: If R(x) tends to a finite limit when x → 1, then the series ∑ n ≥ 1 R f (...

convergent Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a non-convergent series....

vector space), the series ∑ n = 0 ∞ f n ( x ) {\displaystyle \sum _{n=0}^{\infty }f_{n}(x)} is called normally convergent if the series of uniform norms...

its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open...

series and the negative degree power series converge. Furthermore, this convergence will be uniform on compact sets. Finally, the convergent series defines...

summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows...

series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if...

converge absolutely if ∑ | a n | {\textstyle \sum |a_{n}|} is convergent. A convergent series ∑ a n {\textstyle \sum a_{n}} for which ∑ | a n | {\textstyle...

2004, pp. 53–54 Cooker, M.J. (2011). "Fast formulas for slowly convergent alternating series" (PDF). Mathematical Gazette. 95 (533): 218–226. doi:10.1017/S0025557200002928...

Convergent thinking is a term coined by Joy Paul Guilford as the opposite of divergent thinking. It generally means the ability to give the "correct" answer...

carnivorous aliens as food animals. In the notes to his collection Convergent Series, Niven wrote that "Bordered in Black" does not belong to the Known...