haraatmak shreni

ganit mein haraatmak shreni apasaari anant shreni hai:

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

iska naamakaran samaantar shreni ke vyutkram se hua hai. samaantar shreni ke padon ko yahaan har mein likha jaata hai arthaat samaantar shreni ke pad ${\displaystyle a_{i}}$ se sambandhit haraatmak shreni ka pad ${\displaystyle {\frac {1}{a_{i}}}}$ hai.

angreji mein ise haarmonik shreni kehte hain jiski avadhaarana sangeet ki dhun se hua. ek kampanasheel tantu se nikalne vaal adhisvarak (dhun) 1/2, 1/3, 1/4, aadi, tantu ki moolabhoot tarangadairdhya hain. pratham pad ke baad shreni ka pratyek pad apne paas vaale padon ka haraatmak maadhya hota hai.

chaudaveen shataabdi mein nikol oresm nein yeh siddh kiya ki haraatmak shreni apasaari hoti hai lekin yeh parinaam kisi ne naheen dekhe. 17 veen shataabdi mein petro mangoli, johaan barnooli aur Jacob barnooli iski upapatti ki.

aitihaasik drushti se haraatmak anukram ko vaastukaaron mein kuchh prasiddhi mili. yeh vishesh roop se barauk kaal mein hui jab vaastukaaron ne unnayan (oonchaai) aur chhat samatal ke madhya santulan ki sthaapana ke liye upayog kiya tatha charchon aur mahalon mein baahari aur aantarik kala mein sannaadi sambandh sthaapit kiya.^[1]

prathamadrushtaya yeh shreni sahaj naheen lagti kyonki yeh ek apasaari shreni hai balki iska n vaaain pad jab n anant ki or agrasar hai, shoonya ki or agrasar hota hai. haraatmak shreni ka apasaran virodhaabhaas ka ek spasht udaaharan hai. iska ek udaaharan rabar ke fite par keeda hai.^[2] maana ki ek keeda ek meter lambe rabar ke feete par mand gati se chalta hai, pratyek ek minute pashchaat rabar ka feeta samaroop ek meter kheench kar lamba ho jaata hai. yadi keeda ek semi prati minute ki dar se reingata hai, kya keeda feete ke doosare sire par kabhi naheen pahuainga paayega? lekin uttar ekdam viprit "haaain" hoga, n minute pashchaat keede dvaara tay doori aur feete ki kul lambaai ka anupaat nimn hoga:

${\displaystyle {\frac {1}{100}}\sum _{k=1}^{n}{\frac {1}{k}}.}$

kyonki shreni ka maan svechh roop se badhta hai jaise jaise n ka maan badhata hai, at: yeh anupaat ek se adhik bhi hona chaahiye jisse siddh hota hai ki keeda feete ke ant tak pahuainch jaayega. yeh ghatna ghatit hone mein lagne wala samay arthaat n ka maan adhik ho sakta hai, tathaapi, lagbhag e¹⁰⁰ athva 10⁴⁰ se bhi adhik. yaddapi haraatmak shreni apasaari hai at: yeh itna dheemein hota hai.

haraatmak shreni ke apasaari hone ke anek parinaam upalabddh hain jinmein se do yahaaain diye gaye hain.

tulana pareekshan

ise apasaari siddh karne ka pratham tarika ise kisi anya apasaari shreni ke saath tulana karne ka hai

${\displaystyle {\begin{aligned}&1\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{3}}\,+\,{\frac {1}{4}}\;\;+\;\;{\frac {1}{5}}\,+\,{\frac {1}{6}}\,+\,{\frac {1}{7}}\,+\,{\frac {1}{8}}\;\;+\;\;{\frac {1}{9}}\,+\,\cdots \\[12pt]>\;\;\;&1\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{4}}\,+\,{\frac {1}{4}}\;\;+\;\;{\frac {1}{8}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{8}}\;\;+\;\;{\frac {1}{16}}\,+\,\cdots .\end{aligned}}}$

haraatmak shreni ka pratyek pad isse tulana ki gayi shreni ke pratyek pad se bada athva baraabar hai at: haraatmak shreni ka sankalan (yog) bhi dviteeya shreni ke yog se adhik hoga. yaddapi dviteeya shreni ka kul yog anant hai:

${\displaystyle {\begin{aligned}&1+\left({\frac {1}{2}}\right)+\left({\frac {1}{4}}+{\frac {1}{4}}\right)+\left({\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}\right)+\left({\frac {1}{16}}+\cdots +{\frac {1}{16}}\right)+\cdots \\[12pt]=\;\;&1\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{2}}\;\;+\;\;\cdots \;\;=\;\;\infty .\end{aligned}}}$

isse siddh hota hai ki haraatmak shreni ka yog bhi anant hona chaahiye. doosare shabdon mein uparokt tulana se siddh hota hai ki

${\displaystyle \sum _{n=1}^{2^{k}}\,{\frac {1}{n}}\;\geq \;1+{\frac {k}{2}}}$

jahaaain k ek dhanaatmak poornaank hai.

samaakalan pareekshan

yeh siddh karna sambhav hai ki haraatmak shreni ke sankalan ko iski tulanaatmak shreni ke anant samaakal se tulana karne par apasaari praapt hoti hai. vishesh roop se, maana chitr mein pradarshit aayaton mein pradarshit tarkon ko sahi hain. pratyek aayat ki chaudaai ek ikaai aur aur ooainchaai 1/n ikaai hai, at: sabhi aayaton ka kul kshetrafal, haraatmak shreni ke kul yog ke baraabar hoga:

sabhi aayaton ka kul kshetrafal ${\displaystyle =1\,+\,{\frac {1}{2}}\,+\,{\frac {1}{3}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{5}}\,+\,\cdots .}$

yaddapi, vakr y= 1/x ke neeche 1 se anant tak ka kshetrafal nimn prakaar hai:

vakr ke neeche ka kshetrafal ${\displaystyle =\int _{1}^{\infty }{\frac {1}{x}}\,dx\;=\;\infty .}$

choonki yeh kshetrafal poornataya aayat ke andar sthit hai at:

${\displaystyle \sum _{n=1}^{k}\,{\frac {1}{n}}\;>\;\int _{1}^{k+1}{\frac {1}{x}}\,dx\;=\;\ln(k+1).}$

haraatmak shreni bahut mand roop se apasaari hai. udaaharan ke liye, iske pratham 10⁴³ padon ka yog 100 se kam hai.^[3] yeh isliye ki shreni ka aanshik sankalan mein laghuganakeeya vruddhi hoti hai.

${\displaystyle \sum _{n=1}^{k}\,{\frac {1}{n}}\;=\;\ln k+\gamma +\varepsilon _{k}\leq \ln k+1}$

jahaaain ${\displaystyle \gamma }$ oyalar mascheroni niytaank (Euler–Mascheroni constant) ^{(krupaya sahi hindi uchchaaran likhein)} aur ${\displaystyle k}$ ke shoonya ki taraf agrasar hone par ${\displaystyle \varepsilon _{k}}$ ~ ${\displaystyle {\frac {1}{2k}}}$ anant ki or agrasar hota hai. liyonaard oyalar ke anusaar

${\displaystyle \sum _{p{\text{ prime }}}{\frac {1}{p}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+\cdots =\infty .}$

apasaari haraatmak shreni ka nvaaain aanshik sankalan

${\displaystyle H_{n}=\sum _{k=1}^{n}{\frac {1}{k}},\!}$

ko nveen haraatmak sankhya kaha jaata hai.

• sammishr laghuganak
• reemaan parikalpana
• haraatmak shredhi

• hejvinkl, michyel, san. (2001), "haraatmak shreni (Harmonic series)", ensaaiklopeediya of maithamaitiks, springar, aai॰aऍsa॰abee॰aऍna॰ 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=p/h046540
• "The Harmonic Series Diverges Again and Again", The AMATYC Review, 27 (2006), pp. 31–43. Many proofs of divergence of harmonic series.
• erik dablyoo veisateen, maithavarld par haraatmak shreni (Harmonic Series)
• Use Of The Zonal Harmonic Series For Obtaining Numerical Solutions To Electromagnetic Boundary Value Problems; Proceedings Of Thru-The-Earth Electromagnetics Workshop