Thus, the generating function $ \sum _ {h \geq 0 } a _ {h} X ^ {h} $
are given by a generalized power sum $ a _ {h} = a ( h ) = \sum _ {i = 1 } ^ {m} A _ {i} ( h ) \alpha _ {i} ^ {h} $ ($ h = 0,1, \dots $
6 KB (908 words) - 06:04, 12 July 2022
For functions represented by a power series, a majorant is e.g. the sum of a power series with positive coefficients which are not less than the absolute valu
2 KB (302 words) - 10:06, 15 April 2014
is summable by the Hölder method $(H,k)$ to sum $s$ if
...mable to a sum $s$ by the method $(H,k)$, it will also be summable to that sum by the method $(H,k')$ for any $k'>k$. For any $k$ the method $(H,k)$ is eq
2 KB (280 words) - 13:46, 14 February 2020
Consider a complex power series
\sum _ {k = 0 } ^ \infty
4 KB (625 words) - 15:35, 4 June 2020
The formula for the expansion of an arbitrary positive integral power of a [[Binomial|binomial]] in a polynomial arranged in powers of one of the
\sum _ {k = 0 } ^ { m } \left ( \begin{array}{c}
2 KB (262 words) - 08:02, 6 June 2020
\sum _ {m , n = 1 } ^ \infty u _ {mn} ,
S _ {mn} = \sum _ {i = 1 } ^ { m }
7 KB (1,082 words) - 19:36, 5 June 2020
and if the sequence of partial sums of a series $ \sum _ {n = 1 } ^ \infty b _ {n} ( x) $
may take complex values), then the series $ \sum _ {n = 1 } ^ \infty a _ {n} ( x) b _ {n} ( x) $
1 KB (212 words) - 07:38, 1 November 2023
$#C+1 = 41 : ~/encyclopedia/old_files/data/D032/D.0302740 Direct sum
...[[Abelian category|Abelian category]]. In the non-Abelian case the direct sum is usually called the discrete direct product. Let $ \mathfrak A $
4 KB (680 words) - 19:35, 5 June 2020
is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if
...eries. Thus, the series $\sum_{n=0}^\infty z^n$ is $(E,q)$-summable to the sum $1/(1-z)$ in the disc with centre at $-q$ and of radius $q+1$.
2 KB (358 words) - 17:36, 14 February 2020
A power series of the form
\sum _ { n=0 } ^ \infty
3 KB (470 words) - 08:17, 26 March 2023
\sum _ { s=0 } ^ { n }
\sum _ { s=0 } ^ { n-1 }
6 KB (828 words) - 10:58, 29 May 2020
th exterior power $ \wedge ^ {r} V $.
The direct sum of the spaces of skew-symmetric $ r $-
1 KB (155 words) - 19:38, 5 June 2020
\sum _ {k=0} ^ \infty u _ {k} $$
is summable by the Lindelöf summation method to the sum $ s $
3 KB (417 words) - 08:17, 6 January 2024
f ( x + h ) = \sum _ {n = 0 } ^ \infty
P ( x + \xi h ) = \sum _ {\nu = 0 } ^ { m } P _ \nu ( x , h ) \xi ^ \nu ,
6 KB (901 words) - 16:08, 1 April 2020
Abel's theorem on power series: If the power series
S ( z ) \ = \ \sum _ {k = 0} ^ \infty a _ {k} ( z - b ) ^ {k} ,
6 KB (894 words) - 06:14, 26 March 2023
be its power function (cf. [[Power function of a test|Power function of a test]]), which gives for every $ \theta $
the corresponding sequence of power functions $ \{ \beta _ {n} ( \theta ) \} $
7 KB (902 words) - 17:46, 4 June 2020
...r any $x_0\in I$ there is a neighborhood $J$ of $x_0$ and a power series $\sum a_n (x-x_0)^n$ such that
An analytic function is infinitely differentiable and its power expansion coincides with the [[Taylor series]]. Namely, the coefficients $a
6 KB (1,048 words) - 21:19, 14 January 2021
..., where $ \lambda $ is a limit ordinal number and $ n $ is an integer, the sum being understood in the sense of addition of [[Order type|order types]].
...omega $ is the least initial ordinal number. The initial ordinal number of power $ \tau $ is denoted by $ \omega(\tau) $. The set $ \{ \omega(\delta) \mid \
9 KB (1,404 words) - 18:33, 4 December 2017
...uestion must be posed, not what the sum is equal to, but how to define the sum of a divergent series, and he found an approach to the solution of this pro
\sum _ {n = 0 } ^ \infty
4 KB (679 words) - 19:36, 5 June 2020
th symmetric power of $ E $(
th exterior power of the module $ E $(
4 KB (590 words) - 07:45, 7 January 2024