apasaari shreni

ganit mein apasaari shreni ek anant shreni hai jo abhisaari naheen hai, matlab yeh ki shreni ke aanshik yog ka anant anukram ka seemaant maan naheen hota.

yadi ek shreni abhisran karti hai to iska vyaash‍aatikaari pad (nvaaain pad jahaaain n anant ki or agrasar hai.) shoonya ki or agrasar hona chahiye. at: koi bhi shreni jiska vyaash‍aatikaari pad shoonya ki or agrasar naheen hota to vah apasaari hoti hai. tathaapi abhisran ki shart thodi prabal hai: jis shreniyon ka vyaash‍aatikaari pad shoonya ki or agrasar ho vah aavashyak roop se abhisaari naheen hoti. iska ek gananeeya udaaharan nimn haraatmak shreni hai:

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.}$

haraatmak shreni ka apasaran madhyakaaleen ganitjnya nikol oresam dvaara siddh kiya ja chuka hai.

ebal sankalan

yadi λ_n = n, tab hamein ebal sankalan vidhi se praapt hoti hai. yahaaain

${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}\exp(-nx)=\sum _{n=0}^{\infty }a_{n}z^{n},}$

jahaaain z = exp(−x) hai. at: jaise hi x yadi dhanaatmak disha ki or se shoonya ki or agrasar hai to seema ka maan f(x) dhanaatmak vaastavik sankhyaaon ki taraf se z ek (1) ki or agrasar hai to f(z) ki ghaateeya shreni ke liye seema hogi aur ebal sankalan A(s) nimn prakaar paribhaashit hai:

${\displaystyle A(s)=\lim _{z\rightarrow 1^{-}}\sum _{n=0}^{\infty }a_{n}z^{n}}$

ebal sankalan rochak hai kyonki iska sangat hal sisaira-sankalan se adhik prabal hai: A(s) = C_k(s) jab bhi uttaravarti paribhaashit ho.

lindalaaf sankalan

yadi 1 = λ_n = n ln(n), tab (ek se anukraman)

${\displaystyle f(x)=a_{1}+a_{2}2^{-2x}+a_{3}3^{-3x}+\cdots .}$

tab L(s), lindalaaf sankalan (volakauv 2001), jaise x shoonya ki or agrasar ho to ƒ(x) hoga. lindalaaf sankalan ek laabhadaayak vidhi hai jab anya anuprayogon ke madhya ek ghaateeya shreni par laagoo kiya jaata hai.

yadi g(z) chakati ke shoonya ke chaaron or vish‍aaleshanaatmak hai aur at: dhanaatmak trijya ke abhisran sahit maiklaarin shreni G(z) hai, tab mittaag-leffler sitaara (*) mein L(G(z)) = g(z). iske atirikt g(z) ka abhisran is sitaare ke sanhat upasamuchchaya ekaroop hai.

• anant shreni
• abhisaari shreni
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯
• boral sankalan
• oyalar sankalan
• laimbart sankalan

• aarteka, jee॰ e॰; fernaandej, efa॰ ema॰; kaastro, i॰ e॰ (1990), laarj-aardar partabeshan theory end sammeshan methads in kvaantam maikeniks, barlin: springar-veraleg .
• baakar, junior, jee॰ e॰; grevaj-morris, pee॰ (1996), Padé Approximants, Cambridge university press .
• brejinsaki, see॰; jgaliya, ema॰ redivo (1991), Extrapolation Methods. Theory and Practice, uttar haulaind .
• haardi, jee॰ echa॰ (1949), apasaari shreni (Divergent Series), oksaford: klereindaun press, http://www.archive.org/details/divergentseries033523mbp .
• LeGuillou, je॰-see॰; Zinn-Justin, je॰ (1990), Large-Order Behaviour of Perturbation Theory, Amsterdam: uttar haulaind .
• volakauv, aaya॰ aaya॰ (2001), "Lindelöaf summation method", in hejvinkl, michyel, ensaaiklopeediya of maithamaitiks, springar, aai॰aऍsa॰abee॰aऍna॰ 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=l/l058990 .
• jkharauv, e॰ e॰ (2001), "haabil sankalan vidhi (Abel summation method)", in hejvinkl, michyel, ensaaiklopeediya of maithamaitiks, springar, aai॰aऍsa॰abee॰aऍna॰ 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=a/a010170 .