sisaira-sankalan

ganiteeya vishleshan mein, sisairo sankalan saamaanya arthon mein abhisran naheen karne vaale anant sankalan ko maan nirdisht karta hai jo maanak sankalan ke saath sanniptit hota hai yadi vah abhisaari ho. sisaira sankalan shreni ke aanshik sankalan ke samaantar maadhya ke seemaant maan ke roop mein paribhaashit hota hai.

sisaira sankalan ka naamakaran itaalavi vishleshak arnesto sisaira (1859–1906) ke sammaan mein rakha gaya.

maana {a_n} ek anukram hai aur maana

${\displaystyle s_{k}=a_{1}+\cdots +a_{k}}$

shreni ka kvaaain aanshik sankalan hai

${\displaystyle \sum _{n=1}^{\infty }a_{n}.}$

shreni ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ sisaira yog ${\displaystyle A\in \mathbb {R} }$ ke saath sisaira sankalaneeya kahalaati hai yadi iske aanshik sankalanon ke yog ka maadhya ${\displaystyle s_{k}}$, ${\displaystyle A}$ ki or agrasar ho:

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}s_{k}=A.}$

anya shabdon mein, kisi anant shreni ka sisaira sankalan shreni ke pratham n aanshik sankalanon ka samaantar maadhya (ausat) ka seemaant maan hota hai jabki n anant ki or agrasar ho. yeh siddh kiya ja sakta hai ki abhisaari shreni sisaira sankalaneeya hoti hai aur shreni ka kul yog sisaira sankalan ke samaan hota hai. haalaanki, nimnalikhit udaaharan ullikheet karta hai ki shreni apasaari hai lekin sisaira sankalaneeya hai.

maana n ≥ 1 ke liye a_n = (−1)ⁿ⁺¹ hai. arthaat {a_n} ek anukram hai

${\displaystyle 1,-1,1,-1,\ldots .\,}$

aur maana G ek shreni ${\displaystyle \sum _{n=1}^{\infty }a_{n}=1-1+1-1+1-\cdots }$ ko nirupit karta hai.

tab aanshik sankalanon {s_n} ${\displaystyle =\sum _{k=1}^{n}a_{k}}$ ka anukram nimnalikhit hoga

${\displaystyle 1,0,1,0,\ldots ,\,}$

is shreni G ko graandi shreni ke roop mein jaana jaata hai jo abhisaari naheen hai. anya roop mein, {s_n} ke padon ke (aanshik) maadhya anukram {t_n} hai jahaaain

${\displaystyle t_{n}={\frac {1}{n}}\sum _{k=1}^{n}s_{k}}$

nimn prakaar hain

${\displaystyle {\frac {1}{1}},\,{\frac {1}{2}},\,{\frac {2}{3}},\,{\frac {2}{4}},\,{\frac {3}{5}},\,{\frac {3}{6}},\,{\frac {4}{7}},\,{\frac {4}{8}},\,\ldots ,}$

at:

${\displaystyle \lim _{n\to \infty }t_{n}=1/2.}$

isliye shreni G ka sisaira sankalan ka maan 1/2 hai.

anya roop mein, maana n ≥ 1 ke liye a_n = n hai. arthaat {a_n} ek anukram hai.

${\displaystyle 1,2,3,4,\ldots .\,}$

aur maana G ek shreni ${\displaystyle \sum _{n=1}^{\infty }a_{n}=1+2+3+4+5+\cdots }$ ko nirupit karta hai.

tab {s_n} ke aanshik sankalanon ka anukram

${\displaystyle 1,3,6,10,\ldots ,\,}$

hoga aur G ka maan anant ki or apasarit hoga.

{tn } ke aanshik sankalanon ke maadhya ke anukram ke pad nimn prakaar honge:

${\displaystyle {\frac {1}{1}},\,{\frac {4}{2}},\,{\frac {10}{3}},\,{\frac {20}{4}},\,\ldots .}$

at:, yeh anukram G ki tarah anant ki or apasaran karta hai tatha ab G sisaira sankalaneeya naheen hai.

• haabil sankalan sootr
• apasari shreni
• aayalar sankalan
• sisera maadhya
• reej maadhya
• fejar-prameya
• laimbart sankalan

• Shawyer, Bruce; Watson, Bruce (1994), Borel's Methods of Summability: Theory and Applications, Oxford UP, ISBN 0-19-853585-6 .
• titchamaarsh, Edward Charles (1948), fooriye samaakal siddhaant ka ek parichay (Introduction to the theory of Fourier integrals) (2nd san॰), New York: Chelsea Pub. Co. (published 1986), aai॰aऍsa॰abee॰aऍna॰ 978-0-8284-0324-5 .
• vaulakauv, aai॰aaai॰ (2001), "sisaira sankalan vidhi (Cesàaro summation methods)", in hejvinkl, michyel, ensaaiklopeediya of maithamaitiks, springar, aai॰aऍsa॰abee॰aऍna॰ 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=c/c021360
• jayagamund, antoni (1968), trikonamiteeya shrrunkhala (Trigonometric series) (2nd san॰), Cambridge university press (published 1988), aai॰aऍsa॰abee॰aऍna॰ 978-0-521-35885-9 .