U Vikipediyi ye statti pro inshi znachennya cogo termina Suma znachennya Rezultati obchislennyaporDodavannya 1 j dodanok 2 j dodanok sumaVidnimannya zmenshuvane vid yemnik riznicyaMnozhennya 1 j mnozhnik 2 j mnozhnik dobutokDilennya dilene dilnik chastkaDilennya z ostacheyu mod dilene mod dilnik ostachaPidnesennya do stepenyaosnova stepenyapokaznik stepenya stepinObchislennya korenya pokaznik korenya pidkorenevij viraz korinLogarifm log logosnova chislo logarifmSu ma lat summa rezultat operaciyi dodavannya Napriklad u virazi 4 5 99 ye sumoyu a chisla 4 i 5 nazivayutsya dodankami Suma poznachayetsya znakom plyus Dlya poznachennya sumi chleniv poslidovnosti vikoristovuyetsya simvol displaystyle sum velika grecka litera sigma napriklad i 1 N a i a 1 a 2 a N displaystyle sum i 1 N a i a 1 a 2 ldots a N Yaksho poslidovnist neskinchenna to taka suma nazivayetsya chislovim ryadom i poznachayetsya i 1 a i displaystyle sum i 1 infty a i V algebrayichnij viraz mozhut vhoditi chleni znaki yakih napered ne viznacheni Tobto dlya pevnih chleniv virazu vikonuyetsya operaciya dodavannya dlya inshih vidnimannya Tomu viraz zagalnogo viglyadu do yakogo vhodyat operaciyi dodavannya i vidnimannya nazivayut algebrayichnoyu sumoyu Napriklad 5 4 5 4 displaystyle 5 4 5 4 a b a b displaystyle a b a b Zmist 1 Viznachena suma 2 Prikladi 3 Neviznachena suma 4 Formula Nyutona Lejbnica 5 Etimologiya 6 Div takozhViznachena suma RedaguvatiChasto dlya skorochennya sumu z n dodankiv ak ak 1 aN poznachayut velikoyu greckoyu bukvoyu S sigma a k a k 1 a N i k N a i displaystyle a k a k 1 a N sum i k N a i Ce poznachennya nazivayetsya viznachenoyu skinchennyu sumoyu ai po i vid k do N Dlya zruchnosti zamist i k N a i displaystyle sum i k N a i inkoli pishut P i a i displaystyle sum P i a i de P i displaystyle P i deyake vidnoshennya dlya i displaystyle i takim chinom P i a i displaystyle sum P i a i ce skinchenna suma vsih a i displaystyle a i de i Z P i displaystyle i in Z P i Vlastivosti viznachenoyi sumi i k 1 k 2 a i j p 1 p 2 b j i k 1 k 2 j p 1 p 2 a i b j displaystyle left sum i k 1 k 2 a i right left sum j p 1 p 2 b j right sum i k 1 k 2 left sum j p 1 p 2 a i b j right i k 1 k 2 j p 1 p 2 a i j j p 1 p 2 i k 1 k 2 a i j displaystyle sum i k 1 k 2 sum j p 1 p 2 a ij sum j p 1 p 2 sum i k 1 k 2 a ij i k 1 k 2 a i b i i k 1 k 2 a i i k 1 k 2 b i displaystyle sum i k 1 k 2 a i b i sum i k 1 k 2 a i sum i k 1 k 2 b i i k 1 k 2 z a i z i k 1 k 2 a i displaystyle sum i k 1 k 2 z cdot a i z cdot sum i k 1 k 2 a i Prikladi RedaguvatiSuma arifmetichnoyi progresiyi i 0 n a 0 b i n 1 a 0 a n 2 displaystyle sum i 0 n a 0 b cdot i n 1 frac a 0 a n 2 Suma geometrichnoyi progresiyi i 0 n a 0 b i a 0 1 b n 1 1 b displaystyle sum i 0 n a 0 cdot b i a 0 cdot frac 1 b n 1 1 b i 0 n 1 p i p p 1 1 1 p n 1 p 1 n 0 displaystyle sum i 0 n left frac 1 p right i frac p p 1 left 1 frac 1 p n 1 right quad p neq 1 n geq 0 Chomu ce tak i 0 n 1 p i i 0 n 1 1 p i 1 1 1 p n 1 1 1 p p n 1 1 p n 1 p 1 p p n 1 1 p n p 1 p p 1 1 1 p n 1 displaystyle sum i 0 n left frac 1 p right i sum i 0 n 1 cdot frac 1 p i 1 cdot frac 1 left frac 1 p right n 1 1 frac 1 p frac frac p n 1 1 p n 1 frac p 1 p frac p n 1 1 p n p 1 frac p p 1 left 1 frac 1 p n 1 right i 0 n i p i n p n 2 n 1 p n 1 p p 1 2 p 1 displaystyle sum i 0 n ip i frac np n 2 n 1 p n 1 p p 1 2 quad p neq 1 Chomu ce tak Dovedennya i 0 n i p i i 1 n i p i p i 1 n i p i 1 p i 0 n 1 i 1 p i p i 0 n 1 i p i i 0 n 1 p i p i 0 n i p i p n p n p 1 p n 1 p displaystyle sum i 0 n ip i sum i 1 n ip i p cdot sum i 1 n ip i 1 p cdot sum i 0 n 1 i 1 p i p cdot left sum i 0 n 1 ip i sum i 0 n 1 p i right p cdot sum i 0 n ip i p cdot np n p cdot frac 1 p n 1 p Rightarrow 1 p i 0 n i p i n p n 1 1 p p p n 1 1 p i 0 n i p i n p n 2 n 1 p n 1 p 1 p 2 displaystyle Rightarrow 1 p sum i 0 n ip i frac np n 1 1 p p p n 1 1 p Rightarrow sum i 0 n ip i frac np n 2 n 1 p n 1 p 1 p 2 i 0 n p i p 1 i 0 n 1 n i p i n 1 p 1 displaystyle sum i 0 n p i p 1 sum i 0 n 1 n i p i n 1 quad p neq 1 Chomu ce tak bo tak Dovedennya p 1 i 0 n 1 n i p i n 1 p 1 i 0 n n i p i n 1 p 1 n i 0 n p i i 0 n i p i n 1 displaystyle p 1 sum i 0 n 1 n i p i n 1 p 1 sum i 0 n n i p i n 1 p 1 left n cdot sum i 0 n p i sum i 0 n ip i right n 1 p 1 n 1 p n 1 1 p n p n 2 n 1 p n 1 p 1 p 2 n 1 displaystyle p 1 left n cdot frac 1 p n 1 1 p frac np n 2 n 1 p n 1 p 1 p 2 right n 1 n p n 2 n p n p n 1 n n p n 2 n p n 1 p n 1 p p n n p 1 p 1 displaystyle frac np n 2 np np n 1 n np n 2 np n 1 p n 1 p pn n p 1 p 1 p n 1 1 p 1 i 0 n p i displaystyle frac p n 1 1 p 1 sum i 0 n p i Pri p 10 displaystyle p 10 otrimuyemo i 0 n 10 i 9 i 0 n 1 n i 10 i n 1 displaystyle sum i 0 n 10 i 9 cdot sum i 0 n 1 n i 10 i n 1 a ce poslidovnist rivnyan nastupnogo viglyadu 1 9 0 1 11 9 1 2 111 9 12 3 1111 9 123 4 11111 9 1234 5 displaystyle 1 9 cdot 0 1 quad 11 9 cdot 1 2 quad 111 9 cdot 12 3 quad 1111 9 cdot 123 4 quad 11111 9 cdot 1234 5 Neviznachena suma RedaguvatiNeviznachenoyu sumoyu ai po i nazivayetsya taka funkciya f i yaka poznachayetsya i a i displaystyle sum i a i sho i f i 1 f i a i 1 displaystyle forall if i 1 f i a i 1 Formula Nyutona Lejbnica RedaguvatiDokladnishe Teorema Nyutona LejbnicaYaksho znajdena neviznachena suma i a i f i displaystyle sum i a i f i todi i k N a i f N 1 f k displaystyle sum i k N a i f N 1 f k Etimologiya RedaguvatiLatinske slovo summa perekladayetsya yak golovnij punkt sutnist pidsumok Z XV stolittya slovo pochinaye vzhivatisya v suchasnomu sensi z yavlyayetsya diyeslovo pidsumuvati 1489 rik dzherelo Ce slovo proniklo v bagato suchasnih mov v ukrayinsku anglijsku francuzku ta inshi Specialnij simvol dlya poznachennya sumi S pershim vviv Ejler v 1755 roci Yak variant vikoristovuvalasya grecka bukva Sigma S Piznishe zvazhayuchi na zv yazok ponyat pidsumovuvannya ta integruvannya S takozh vikoristovuvali dlya poznachennya operaciyi integruvannya Div takozh Redaguvati Portal Matematika Dobutok Kontrolna suma Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Suma amp oldid 38653637
