Amxg aviv Ifiite Series BDwbU f~wgkv Abyμg I Amxg avivi g a GKwU cöz ÿ m úk i q Q Abyμ gi c` jvi g a MvwYwZK wpý e evi K i Amxg aviv cviqv hvq BDwb Ui D Ïk GB BDwbU k l Avcwb- Abyμg Kx Zv e vl v Ki Z cvi eb, Abyμ gi köwy web vm Ki Z cvi eb, Amxg aviv wpwýz Ki Z cvi eb, Amxg YvËi avivi mgwó wby qi kz e vl v Ki Z cvi eb, Amxg YvËi avivi mgwó wby q Ki Z cvi eb, cšbtcywbk `kwgk fmœvsk K AbšÍ YvËi avivq cökvk Ki Z cvi eb cšbtcywbk `kwgk fmœvsk K mvaviy fmœvs k iƒcvšíi Ki Z cvi eb BDwbU mgvwßi mgq BDwbU mgvwßi m ev P mgq w`b GB BDwb Ui cvvmg~ cvv.: Abyμg I Amxg aviv cvv.: YvËi avivi mgwó cvv.: cšbtcywbk `kwg Ki mvaviy fmœvs k iƒcvšíi
I cb zj GmGmwm cövmövg cvv. Abyμg I Amxg aviv cvvwfwëk D Ïk GB cvv k l Avcwb- Abyμg Kx Zv e vl v Ki Z cvi eb, aviv Kx Zv wjl Z cvi eb, Amxg aviv Kx Zv e vl v Ki Z cvi eb, Amxg avivi mgvavb Ki Z cvi eb gyl kã Abyμg, Amxg aviv g~jcvv- Abyμg (Sequece): wb Pi m úk wu jÿ Kiæb Ñ 4 6...... 4 6 8 0... GLv b cö Z K N Gi Rb Gi wø Y Gi mv _ m úwk Z h Zz ^vfvwek msl vi mu, N = {,,, 4,, 6,...} BZ vw` _ K GKwU wbq gi gva g Gi wø Y msl vi mu = {, 4, 6, 8, 0,,...} cviqv hvq G wø Y ev wøzxqevi cövß msl v jvi mu GKwU Abyμg myzivs KZK jv ivwk GKUv we kl wbq g μgvš^ q Ggbfv e mw¾z q h, cö Z KwU ivwk Gi c~e c` I c ii c `i mv _ wkfv e m úwk Z Zv Rvbv hvq- ZvB Abyμg (sequece) Avevi Dc ii m úk wu K dvskb e j Ges f() = jlv q GB Abyμ gi mvaviy c` Ges AbyμgwU K mvaviy c `i mvv h jlvi c wz jv {} hlv b =,,,... ev, Avevi AbyμgwU Z Ô...Õ Øviv Giƒc AšÍxbfv e Pj Z _vk e wb ` k K i Zv j Abyμ gi mvaviy AvKvi jv, a, a, a,... a GLv b a, a, a K h_vμ g AbyμgwUi cö_g, wøzxq I Z Zxq c` (Term) e j Ges a K Zgc` ev mvaviy c` e j a - K a mve cov q Abyμ gi ivwk jv μgvš^ q e w c Z _vk j Zv K EaŸ μwgk Ges «vm c Z _vk j Zv K wb œ μwgk Abyμg e j c` Abymv i Abyμg `yb cökvi h_v: () mvší/aší/mmxg Abyμg () AmvšÍ/AbšÍ/Amxg Abyμg h Abyμ gi c` msl v wbw` ó A_v r MbYv K i kl Kiv hvq, Zv K mmxg ev mvší Abyμg Ges h Abyμ gi c` msl v MbYv K i kl Kiv hvq bv, Zv K AbšÍ ev Amxg Abyμg e j hgb:,,, 7,..., GKwU mmxg Abyμg Avevi,, 4, 6, 8,... GKwU Amxg Abyμg wb Pi KZK jv Abyμg `Iqv jv cö Z K N Gi Rb,,,..., 4,...,,,,, 6..., (7 ),... 8 evsjv `k Dš y³ wek we` vjq
D PZi MwYZ BDwbU,, 0, 7, 6 wkÿv_ xi KvR...,,.... wb Pi Abyμg jvi mvaviy c` wby q Kiæb: 7 (i),,,,... ii),, 7,,... 4 6 8 4 4 4 4 (iii) 4,,,,,... 7 8. cö`ë mvaviy c` Z wb Pi Abyμg jv wjlyb: (i) (-) (ii) (iii) (iv) cos aviv (Series): Kv bv Abyμ gi c` jv cici Ò+Ó wpý Øviv hy³ Kiv j, GKwU aviv ev köwy cviqv hvq hgb: (i) 4 + 7 + 0 + +... (ii) + 6 + + 4 +... GLv b (i) bs avivwui cöwzwu c` Zvi c~e ezx c `i mv _ ÒÓ hvm K i Ges (ii) bs avivi cöwzwu c` Zvi c~e ezx c` K Øviv Y K i MwVZ q Q úôzb avivi MVb MYbv _vk j Avgiv avivwui h Kv bv msl K c` wby q Ki Z cvwi avivi c `i msl v MYbvZxZ j, Zv K Amxg aviv Ges c` msl v wbw` ó j, G K mmxg aviv e j Amxg ev AbšÍ aviv (Ifiite Series): U, U, U,..., U,...ev Íe msl vi GKwU Abyμg j, U + U + U +...+ U +... K ev Íe msl vi GKwU Amxg ev AbšÍ aviv (Ifiite Series) Ges U K GB avivi -Zg c` ejv q wb Pi cö Z KwU Amxg aviv: (K) + + +... (L) + +... (M) + + 4 + 8 +... cö Z KwU Amxg avivi AvswkK mgwó (Partial Sum) wby q Kiv hvq g AvswkK mgwó S = U q AvswkK mgwó S = U + U q Avswkm mvgwó S = U + U + U +... Zg AvswkK mgwó S = U + U + U +... + U. A_v r Kv bv Amxg avivi Zg AvswkK mgwó Q avivwui cö_g msl K ( N) c `i mgwó D`vib : cö`ë Amxg aviv `ybwui AvswkK mgwó wby q Kiæb (K) + + + 4 +... (L) + + Ñ... mgvavb: (K) + + + 4 +... avivwui g AvswkK mgwó S = q AvswkK mgwó S = + = q AvswkK mgwó S = + + = 6 Zg AvswkK mgwó S = + + +... = (L) + +... avivwui g AvswkK mgwó S = q AvswkK mgwó S = = 0 Amxg aviv
I cb zj GmGmwm cövmövg q AvswkK mgwó S = + = 4_ AvswkK mgwó S 4 = + = 0 Gfv e AMÖmi j `Lv hvq h, we Rvo j Zg AvswkK mgwó S = Ges Rvo j, -Zg AvswkK mgwó S = 0 ÒLÓ bs avivwu Z Ggb Kv bv wbw` ó msl v cviqv hvq bv hv K avivwui mgwó ejv q cvv. YvËi avivi mgwó cvvwfwëk D Ïk GB cvv k l Avcwb- YvËi avivi mvaviy AbycvZ wby q Ki Z cvi eb, YvËi avivi mgwó wby q Ki Z cvi eb gyl kã mvaviy AbycvZ, YvËi aviv g~jcvv- YvËi aviv (Defiitio of Geometrie Progressio): Kv bv avivi AšÍM Z c`mg~ i h Kv bv GKwU c` I Zvi c~e ezx c `i AbycvZ me Î GKB ev mgvb _vk j D³ aviv K YvËi aviv (Geometric progressio) ev ms ÿ c G.P. e j hgb: +... GKwU Amxg YvËi aviv 4 8 mvaviy AbycvZ (Commo Ratio): YvËi avivi h Kv bv c` I Zrc~e ezx c `i AbycvZ K mvaviy AbycvZ ev 6 ms ÿ c C.R. ejv q hgb: + 6 + + 4 +... GKwU Amxg YvËi aviv GLv b, 6 4 Ges = myzivs C. R = YvËi cömgb ev avivi -Zg c` ( klc`) wby q: g b Kiæb, Kv bv YvËi cömgb ev avivi g c` a Ges mvaviy AbycvZ r Zv j, cömgbwu a, ar, ar, ar,... cö_g c` = a = ar 0 = ar wøzxq c` = ar = ar - Z Zxq c` = ar = ar -... Zg c` ar - = ar - YvËi avivi msl K c `i mgwó wby q g b Kiæb, YvËi avivi cö_g c` a mvaviy AbycvZ r Ges msl K c `i mgwó S j S = a + ar + ar + ar +... + ar... (i) (S ) r = ar + ar + ar +... + a r.(ii) [r Øviv Y K i] (i) bs Z (ii) bs mgxkiy we qvm K i cvb- 00 evsjv `k Dš y³ wek we` vjq
D PZi MwYZ BDwbU S (S )r = a ar ev, S ( r) = a ( r ) [hlb r < ] = a r r..(iii) Avevi, (ii) bs Z (i) bs mgxkiy we qvm K i cvb Ñ ( )r = ar a ev, S (r ) = a (r ) [hlb r > ] S r = a....(iv) r g b ivlvi welq: (i) r < j, A_v r - < r < j, Gi gvb e w Ki j r Gi gvb «vm cvq Ges - K h _ó eo K i r Gi gvb K h _ó QvU Kiv hvq A_v r 0 Gi h _ó KvQvKvwQ Avbv hvq G _ K ejv hvq h, r < j r Gi cövšíxq gvb 0 q Ges S Gi cövšíxq gvb S = a myzivs a + ar + ar +... AbšÍ avivi mgwó, = r a ar = a r r r r a a r (ii) r > j, A_v r r > A_ev r < j, Gi gvb e w Ki j r Gi gvb ew cvq Ges - K h _ó eo K i r Gi gvb h _ó eo Kiv hvq G _ K `Lv hvq h, Ggb Kv bv wbw` ó msl v S cviqv hvq bv, hv K S Gi cövšíxq gvb aiv hvq A_v r G ÿ Î Amxg avivwui Kv bv mgwó bb (iii) r = j, S Gi Kv bv cövšíxq gvb cviqv hvq bv Kbbv, Rvo msl v j ( ) = Ges we Rvo msl v j () = G ÿ Î avivwu a a + a a + a a +... e myzivs GB Amxg avivi Kv bv mgwó bb gšíe : Amxg YvËi avivi mgwó (hw` _v K) K S wj L cökvk Kiv q Ges G K avivwui AmxgZK mgwó (Sum up to ifiity) ejv q a A_v r a + ar + ar + ar +... AmxgZK mgwó S = hlb r < r wkÿv_x i KvR GKwU YvËi avivi g c` a I mvaviy AbycvZ r `Iqv Av Q avivwu wjlyb Ges hw` Gi AmxgZK mgwó _v K Z e Zv wby q Kiæb: (i) a =, r = (ii) a =, r =, (iii) a =, r = (iv) a = 8, r = 0 D`viY : wb Pi Amxg YvËi avivi AmxgZK mgwó (hw` _v K) wby q Kiæb: () 4 8... () + 0. + 0.0 + 0.00 +... 4 () + +... (4) +... mgvavb: () GLv b, avivwui cö_g c` a = Ges avivwui mvaviy AbycvZ r = = = < Amxg aviv 0
I cb zj GmGmwm cövmövg avivwui AmxgZK mgwó, S = a r = = 0. () GLv b, cö_g c`, a = Ges mvaviy AbycvZ, r = 0 avivwui AmxgZK mgwó S = = () GLv b, avivwui cö_g c` a = Ges mvaviy AbycvZ r = avivwui AmxgZK mgwó, S = = A_v r S = (4) GLv b, avivwui cö_g c` a = Ges mvaviy AbycvZ r = = 7 = 7 = avivwui AmxgZK mgwó, 0 evsjv `k Dš y³ wek we` vjq
D PZi MwYZ BDwbU cvv. cšbtcywbk `kwg Ki mvaviy fmœvs k iƒcvšíi cvvwfwëk D Ïk GB cvv k l Avcwb- cšbtcywbk `kwgk Z mvaviy fmœvs k iƒcvšíi Ki Z cvi eb, cšbtcywbk `kwgk fmœvsk K AbšÍ YvËi avivq cökvk Ki Z cvi eb gyl kã cšbtcywbk, mvaviy fmœvsk g~jcvv- cšbtcywbk `kwg Ki mvaviy fmœvs k iƒcvšíi D`viY : cšbtcywbk `kwgk jv g~j`xq fmœvs k cökvk Kiæb: (K) 0. (L) 0. 7 (M). 7 (N). mgvavb: (K) 0. = 0.... = 0. + 0.0 + 0.00 +....0 Bv GKwU Amxg YvËi aviv, hvi g c`, a = 0. Ges mvaviy AbycvZ, r = 0. 0. 0. = a 0. 0. r 0. 0. (L) 0. 7 = 0.777... = 0.7+0.007 + 0.00007 +... Bv GKwU Amxg YvËvi aviv, hvi g c` a = 0.7 Ges mvaviy AbycvZ, r = 0. 0 0. 7 = a 0.7 0.7 r 0.0 0.. =.777... (M) 7 0.007 0.7 = + (0.7 + 0.007 + 0.00007 +...) GLv b eübxi fz ii avivwui GKwU Amxg YvËi aviv hvi g c` a = 0.7 Ges mvaviy AbycvZ r = 0.007 0.0 0.7. 7 = + = +. =. +. = + =. (N). = + (0. + 0.000 + 0.000000 +...) GLv b eübxi fz ii avivwu GKwU Amxg YvËi aviv hvi g c`, a = 0. Ges mvaviy AbycvZ, 0.000 r = = 0.00 0.. = + = +. = +... = + = 7 = Amxg aviv 0
I cb zj GmGmwm cövmövg cv VvËi g~j vqb.-.,,, 7 AbyμgwUi 0 Zg c` KvbwU? (K) (L) (M) 7 (N).,,... AbyμgwUi PZz_ c` KZ?. (K) (L) (M) (N) 0 7 8 7, AbyμgwUi Zg c` KvbwU? (L) () - (K) (M) ( ) - (N) () -. 4.... GKwU Amxg YvËi aviv j, Zg c` KZ? x x x (K) (L) (M) (N) 4. Kv bv Abyμ gi Zg c`, hlv b N j Z Zxq c` KvbwU? (K) (L) (M) (N) 6 0 6. Kv bv Abyμ gi Zg c` mlv b N j 0 Zg c` KvbwU? (K) 0 (L) (M) (N) 4 4 cv ki avivwu jÿ Kiæb Ges (7-) b ^i cö kœi DËi w`b: 4,,... 7 avivwui 0 Zg c` KvbwU? 4 4 4 (K) (L) (M) 0 8. avivwui cö_g 4 c `i mgwó KZ? 484 60 0 0 (K) (L) (M) (N) 7. avivwui AmxgZK mgwó- (K) 0 (L) (M) 6 (N) 7 0. Kv bv Abyμ gi Zg c` jv U = 4 (N) (K) U 0, U 00, U 000 wby q Kiæb (L) U < 0 j, Gi gvb wkiƒc e Zv wby q Kiæb (M) U Gi cövšíxq gvb, hlb h _ó eo q, m ú K Kx ejv hvq?. MvwYwZK Av iv c wzi mvv h `Lvb h, r jv YvËvi aviv a + ar + ar +... Gi Zg AvswkK mgwó S = a r r. cö`ë Amxg YvËi avivi (AmxgZK) mgwó hw` _v K Z e Zv wby q Kiæb: (K)... (L) 7... 7 7 7 7 4 (M) + 0. + 0.00 +... (N) + + 4 + 8 +... 04 evsjv `k Dš y³ wek we` vjq
D PZi MwYZ BDwbU (O) + 0. + 0.00 +.... Pj Ki Dci Kx kz Av ivc Ki j wb œv³ aviv jvi (AmxgZK) mgwó _vk e Ges mb mgwó wby q Kiæb: (K)... (L) (x + ) - + (x +) - + (x + ) - +... a a a 4. wb Pi cšbtcywbk `kwgk jv K g~j`xq fmœvs k cökvk Kiæb (K) 0. (L) 0.0 (M).0 7 (N).0 (O) 6.40 (P). (Q).. wb Pi aviv jvi cö_g msl K c `i hvmdj wby q Kiæb Ges G `i AmxgZK mgwó _vk j Zv wby q Kiæb bv _vk j e vl v cö`vb Kiæb (K) 4 + 44 + 444 +... (L) + + +... (M) a + aa + aaa +... 6. wb Pi avivwu jÿ Kiæb:... y y y (K) y = j avivwu wby q Kiæb Ges cövß avivwui mvaviy AbycvZ Kiæb (M) K bs G cövß avivwui 0 Zg c` Ges g 0wU c `i mgwó wby q Kiæb (M) cö`ë avivwu y Gi Dci Kx kz Av ivc Ki j avivwui AmxgZK mgwó _vk e Ges mgwó wby q Kiæb Amxg aviv 0
I cb zj GmGmwm cövmövg DËigvjv- cv VvËi g~j vqb. () N () K () M (4) K () M (6) K (7) L (8) L () M (0) (K),, (L) < 0 (M) h _ó eo j U cövšíxq gvb k~b q- 0 00 000 400 0 () (K) ; (L) (M),(N) mgwó bb, (O) 0 (K) kz a < A_ev a > 0 ; mgwó = a (L) kz x < A_ev x > 0; mgwó = x 0 6 84 7 40 4 4 (K), (L), (M), (N), (O), (P), (Q) 6 7 (K) 40 4 0 ; AmxgZK mgwó bb (L) 0 0 ; AmxgZK mgwó bb 8 8 (M) 0 a 0 a ; AmxgZK mgwó bb 8 06 evsjv `k Dš y³ wek we` vjq