Sum function of a series

The summing function of a row is a function that each row $U$ ${\ displaystyle U}$ $U$ matches a certain number $s(U)$ ${\ displaystyle s (U)}$ ${\ displaystyle s (U)}$ . An example of a summing function is $\lim _{n\to \infty }\sum _{k=1}^{n}u_{k}$ ${\ displaystyle \ lim _ {n \ to \ infty} \ sum _ {k = 1} ^ {n} u_ {k}}$ ${\ displaystyle \ lim _ {n \ to \ infty} \ sum _ {k = 1} ^ {n} u_ {k}}$ . This function is defined on the set of all convergent series and its value is equal to the sum of the series . So defined summing function is called $s_{0}$ ${\ displaystyle s_ {0}}$ ${\ displaystyle s_ {0}}$ . For ease of use, summing functions should have regularity properties (if $U$ ${\ displaystyle U}$ $U$ is a convergent series, then the summing function $s(U)$ ${\ displaystyle s (U)}$ ${\ displaystyle s (U)}$ must exist and be equal $s_{0}(U)$ ${\ displaystyle s_ {0} (U)}$ ${\ displaystyle s_ {0} (U)}$ ), and linearity (for any two rows $U$ ${\ displaystyle U}$ $U$ and $V$ ${\ displaystyle V}$ $V$ and numbers $a$ ${\ displaystyle a}$ $a$ and $b$ ${\ displaystyle b}$ $b$ from the existence of values $s(U)$ ${\ displaystyle s (U)}$ ${\ displaystyle s (U)}$ and $s(V)$ ${\ displaystyle s (V)}$ ${\ displaystyle s (V)}$ the existence of meaning follows $s(aU+bV)$ ${\ displaystyle s (aU + bV)}$ ${\ displaystyle s (aU + bV)}$ and equality $s(aU+bV)=as(U)+bs(V)$ ${\ displaystyle s (aU + bV) = as (U) + bs (V)}$ ${\ displaystyle s (aU + bV) = as (U) + bs (V)}$ ) ^[1] .

Examples

The summing Poisson-Abel function is the function defined by the equality $s_{p}=\lim _{x\to 1-0}\lim _{n\to \infty }\sum _{k=1}^{n}u_{k}x^{k}$ ${\ displaystyle s_ {p} = \ lim _ {x \ to 1-0} \ lim _ {n \ to \ infty} \ sum _ {k = 1} ^ {n} u_ {k} x ^ {k} }$ $s_{p}=\lim _{x\to 1-0}\lim _{n\to \infty }\sum _{k=1}^{n}u_{k}x^{k}$ . The summing Poisson-Abel function is regular and linear ^[2] .

Notes

↑ Vorobyov, 1986 , p. 285.
↑ Vorobyov, 1986 , p. 289.

Literature

Vorobiev N.N. Series Theory. - M .: Nauka, 1986 .-- 408 p.

[_628611c1069f2ff4-1] Vorobyov, 1986 , p. 285.

[_628611c1069f2ff8-2] Vorobyov, 1986 , p. 289.

Sum function of a series

Examples

Notes

Literature

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