V matematici garmonichnim ryadom nazivayetsya neskinchennij rozbizhnij ryad k 1 1 k 1 1 2 1 3 1 4 displaystyle sum k 1 infty frac 1 k 1 frac 1 2 frac 1 3 frac 1 4 cdots Zmist 1 Obchislennya 1 1 Deyaki znachennya chastkovih sum 2 Rozbizhnist ryadu 2 1 Dovedennya 1 2 2 Dovedennya 2 2 3 Dovedennya 3 3 Pov yazani ryadi 3 1 Znakoperemizhnij garmonichnij ryad 4 Div takozh 5 Literatura 6 ZnoskiObchislennya Redaguvatin displaystyle n nbsp noyu chastkovoyu sumoyu s n displaystyle s n nbsp garmonichnogo ryadu nazivayetsya n displaystyle n nbsp ne garmonichne chislo s n k 1 n 1 k 1 1 2 1 3 1 4 1 n displaystyle s n sum k 1 n frac 1 k 1 frac 1 2 frac 1 3 frac 1 4 cdots frac 1 n nbsp Deyaki znachennya chastkovih sum Redaguvati s 1 1 s 2 3 2 1 5 s 3 11 6 1 833 s 4 25 12 2 083 displaystyle begin matrix s 1 amp amp 1 s 2 amp amp frac 3 2 amp amp 1 5 s 3 amp amp frac 11 6 amp approx amp 1 833 s 4 amp amp frac 25 12 amp approx amp 2 083 end matrix nbsp s 5 137 60 2 283 s 6 49 20 2 45 s 7 363 140 2 593 s 8 761 280 2 718 displaystyle begin matrix s 5 amp amp frac 137 60 amp approx amp 2 283 s 6 amp amp frac 49 20 amp amp 2 45 s 7 amp amp frac 363 140 amp approx amp 2 593 s 8 amp amp frac 761 280 amp approx amp 2 718 end matrix nbsp Rozbizhnist ryadu RedaguvatiGarmonichnij ryad rozbizhnij shopravda rozbizhnist ye duzhe povilnoyu dlya togo shob chastkova suma perevishila 100 neobhidno blizko 1043 elementiv ryadu Dovedennya 1 Redaguvati Rozbizhnist ryadu mozhna dovesti pogrupuvavshi dodanki tak k 1 1 k 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 gt 1 1 2 1 4 1 4 1 8 1 8 1 8 1 8 1 16 1 1 2 1 2 1 2 1 2 displaystyle begin aligned sum k 1 infty frac 1 k amp 1 left frac 1 2 right left frac 1 3 frac 1 4 right left frac 1 5 frac 1 6 frac 1 7 frac 1 8 right left frac 1 9 cdots right cdots amp gt 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 right cdots amp 1 frac 1 2 quad frac 1 2 quad qquad quad frac 1 2 qquad quad quad frac 1 2 quad cdots end aligned nbsp Ostannij ryad ochevidno rozbizhnij sho dovodit tverdzhennya Dovedennya 2 Redaguvati Pripustimo sho garmonichnij ryad zbizhnij i jogo suma rivna S displaystyle S nbsp k 1 1 k 1 1 2 1 3 1 4 S displaystyle sum k 1 infty frac 1 k 1 frac 1 2 frac 1 3 frac 1 4 cdots S nbsp Todi peregrupuvavshi dodanki oderzhimo S 1 1 3 1 5 1 7 1 2 1 4 1 6 1 8 displaystyle S left 1 frac 1 3 frac 1 5 frac 1 7 cdots right left frac 1 2 frac 1 4 frac 1 6 frac 1 8 cdots right nbsp Vinesemo iz drugih duzhok 1 2 displaystyle tfrac 1 2 nbsp S 1 1 3 1 5 1 7 1 2 1 1 2 1 3 1 4 displaystyle S left 1 frac 1 3 frac 1 5 frac 1 7 cdots right frac 1 2 left 1 frac 1 2 frac 1 3 frac 1 4 cdots right nbsp Zaminimo viraz v drugih duzhkah na S displaystyle S nbsp S 1 1 3 1 5 1 7 1 2 S displaystyle S left 1 frac 1 3 frac 1 5 frac 1 7 cdots right frac 1 2 S nbsp Perenesemo 1 2 S displaystyle tfrac 1 2 S nbsp v livu chastinu 1 2 S 1 1 3 1 5 1 7 displaystyle frac 1 2 S left 1 frac 1 3 frac 1 5 frac 1 7 cdots right nbsp Zaminivshi S displaystyle S nbsp sumoyu ryadu oderzhimo 1 2 1 4 1 6 1 8 1 1 3 1 5 1 7 displaystyle frac 1 2 frac 1 4 frac 1 6 frac 1 8 cdots 1 frac 1 3 frac 1 5 frac 1 7 cdots nbsp Cya rivnist hibna oskilki odinicya bilsha odniyeyi drugoyi odna tretya bilshe odniyeyi chetvertoyi i tak dali Takim chinom pripushennya pro zbizhnist ryadu privelo do superechnosti Dovedennya 3 Redaguvati Na pochatok zapishemo sumu geometrichnoyi progresiyi 1 1 x 1 x x 2 x 3 x 4 displaystyle frac 1 1 x 1 x x 2 x 3 x 4 nbsp de x lt 1 Vizmemo integral z oboh storin vnaslidok chogo oderzhimo ln 1 x x x 2 2 x 3 3 displaystyle ln 1 x x frac x 2 2 frac x 3 3 nbsp Perejshovshi do granici pri x 1 displaystyle x rightarrow 1 nbsp oderzhuyemo rivnist lim x 1 ln 1 x 1 1 2 1 3 1 4 n 1 1 n displaystyle lim x to 1 ln 1 x 1 frac 1 2 frac 1 3 frac 1 4 sum n 1 infty frac 1 n nbsp Oskilki lim x 1 ln 1 x displaystyle lim x to 1 ln 1 x infty infty nbsp to takozh maye misce n 1 1 n displaystyle sum n 1 infty frac 1 n infty nbsp Tobto garmonichnij ryad ye rozbizhnim Pov yazani ryadi RedaguvatiCej rozdil potrebuye dopovnennya Znakoperemizhnij garmonichnij ryad Redaguvati nbsp Pershi 14 chastkovih sum znakoperemizhnogo garmonijnogo ryadu chorni vidrizki zbigayutsya do naturalnogo logarifmu 2 chervona pryama Ryad n 1 1 n 1 n 1 1 2 1 3 1 4 1 5 displaystyle sum n 1 infty frac 1 n 1 n 1 frac 1 2 frac 1 3 frac 1 4 frac 1 5 cdots nbsp nazivayetsya znakoperemizhnim garmonichnim ryadom Vin umovno zbizhnij za teoremoyu Lejbnica ale ne absolyutno zbizhnij Jogo suma logarifm vid 2 en 1 Vikoristannya znakiv sho cherguyutsya z lishe neparnimi znamennikami dast pov yazanij ryad Lejbnica dlya znahodzhennya p 2 n 0 1 n 2 n 1 1 1 3 1 5 1 7 p 4 displaystyle sum n 0 infty frac 1 n 2n 1 1 frac 1 3 frac 1 5 frac 1 7 cdots frac pi 4 nbsp Div takozh RedaguvatiGarmonichne chislo Konstanta Majsselya MertensaLiteratura RedaguvatiFihtengolc G M Kurs differencialnogo i integralnogo ischisleniya Moskva Nauka 1964 T 2 800 s ros Garmonichnij ryad Visha matematika v prikladah i zadachah Klepko V Yu Golec V L 2 ge vidannya K Centr uchbovoyi literaturi 2009 S 497 594 s Znoski Redaguvati Freniche Francisco J 2010 On Riemann s rearrangement theorem for the alternating harmonic series The American Mathematical Monthly 117 5 442 448 JSTOR 10 4169 000298910x485969 MR 2663251 doi 10 4169 000298910X485969 Soddy F 1943 The three infinite harmonic series and their sums with topical reference to the Newton and Leibniz series for p displaystyle pi nbsp Proceedings of the Royal Society 182 113 129 MR 9207 doi 10 1098 rspa 1943 0026 Otrimano z https uk wikipedia org w index php title Garmonichnij ryad amp oldid 40291120
