อน, กรมเรขาคณ, ในทางคณ, ตศาสตร, เป, นอน, กรมท, พจน, างๆ, กสร, างข, นโดยการค, ณพจน, อนหน, าด, วยค, าคงต, วค, าหน, นค, อมาจากลำด, บเรขาคณ, วอย, างเช, displaystyle, over, over, over, over, cdots, infty, over, และโดยท, วไป, displaystyle, infty, จะเป, นอน, กรมล, เข. inthangkhnitsastr xnukrmerkhakhnit epnxnukrmthiphcntang thuksrangkhunodykarkhunphcnkxnhnadwykhakhngtwkhahnung nnkhuxmacakladberkhakhnit twxyangechn 1 1 2 1 4 1 8 1 16 n 0 1 2 n displaystyle 1 1 over 2 1 over 4 1 over 8 1 over 16 cdots sum n 0 infty 1 over 2 n dd aelaodythwip xnukrmerkhakhnit n 0 a r n displaystyle sum n 0 infty ar n dd caepnxnukrmluekhaktxemux r lt 1 displaystyle r lt 1 enuxha 1 phlrwm 1 1 phlrwmyxy 1 2 phlrwmthnghmd 2 karnaipich 2 1 thsniymsa 2 2 xnukrmkalngphlrwm aekikhphlrwmyxy aekikh phlrwmyxykhxng n phcnaerkkhux k 0 n a r k a r 0 a r 1 a r 2 a r 3 a r n displaystyle sum k 0 n ar k ar 0 ar 1 ar 2 ar 3 cdots ar n dd khunthngsxngkhangkhxngsmkardwy 1 r displaystyle 1 r id 1 r k 0 n a r k 1 r a a r a r 2 a r n a a r a r 2 a r n r a a r a r 2 a r n a a r a r 2 a r n a r a r 2 a r 3 a r n 1 a a r n 1 displaystyle begin aligned 1 r sum k 0 n ar k amp 1 r a ar ar 2 ar n amp a ar ar 2 ar n r a ar ar 2 ar n amp a cancel ar cancel ar 2 cancel ar n cancel ar cancel ar 2 cancel ar 3 ar n 1 amp a ar n 1 end aligned dd sungphcnxun catdknhayiphmd cdrupaebbihm caidsutrsahrbkhanwnphlrwm odythi r 1 k 0 n a r k a r n 1 1 r 1 displaystyle sum k 0 n ar k frac a r n 1 1 r 1 dd dngnnkrnithwipkhxngsutrnikhux k m n a r k a r n 1 r m r 1 displaystyle sum k m n ar k frac a r n 1 r m r 1 dd sahrbxnukrmerkhakhnitthimiaetelkhchikalngkhxng r epncanwnkhu khunthngsxngkhangdwy 1 r 2 displaystyle 1 r 2 1 r 2 k 0 n a r 2 k a a r 2 n 2 displaystyle 1 r 2 sum k 0 n ar 2k a ar 2n 2 dd caidsutr k 0 n a r 2 k a 1 r 2 n 2 1 r 2 displaystyle sum k 0 n ar 2k frac a 1 r 2n 2 1 r 2 dd swnelkhchikalngkhxng r thimiaetcanwnkhi 1 r 2 k 0 n a r 2 k 1 a r a r 2 n 3 displaystyle 1 r 2 sum k 0 n ar 2k 1 ar ar 2n 3 dd caidsutr k 0 n a r 2 k 1 a r 1 r 2 n 2 1 r 2 displaystyle sum k 0 n ar 2k 1 frac ar 1 r 2n 2 1 r 2 dd phlrwmthnghmd aekikh samarthkhanwnidcaksutrkhxngphlrwmcakd k 0 a r k lim n k 0 n a r k lim n a 1 r n 1 1 r lim n a 1 r lim n a r n 1 1 r displaystyle sum k 0 infty ar k lim n to infty sum k 0 n ar k lim n to infty frac a 1 r n 1 1 r lim n to infty frac a 1 r lim n to infty frac ar n 1 1 r dd sung r k displaystyle r k camikhaekhaikl 0 emux k mikhaekhaiklxnntktxemux r lt 1 displaystyle r lt 1 dngnn k 0 a r k a 1 r 0 a 1 r displaystyle sum k 0 infty ar k frac a 1 r 0 frac a 1 r dd sahrbxnukrmerkhakhnitthimiaetelkhchikalngkhxng r epncanwnkhu caidsutr k 0 a r 2 k a 1 r 2 displaystyle sum k 0 infty ar 2k frac a 1 r 2 dd swnelkhchikalngkhxng r thimiaetcanwnkhi caidsutr k 0 a r 2 k 1 a r 1 r 2 displaystyle sum k 0 infty ar 2k 1 frac ar 1 r 2 dd odythisutrthnghmddanbncaichidemux r lt 1 displaystyle r lt 1 ethann nxkehnuxcaknicaepnxnukrmluxxkkarnaipich aekikhthsniymsa aekikh sutrphlrwmkhxngxnukrmerkhakhnitichekhiynthsniymsaepnessswnid odytwxyangechn 0 121212 ekhiynidepnxnukrmerkhakhnitthimi a 12 100 aela r 1 100 dngni0 121212 0 12 0 0012 0 000012 12 100 12 10000 12 1000000 12 100 12 100 1 100 12 100 1 100 2 12 100 1 1 100 12 100 99 100 12 99 4 33 displaystyle begin aligned 0 121212 amp 0 12 0 0012 0 000012 amp frac 12 100 frac 12 10000 frac 12 1000000 amp frac 12 100 frac 12 100 left frac 1 100 right frac 12 100 left frac 1 100 right 2 amp frac 12 100 1 1 100 frac 12 100 99 100 frac 12 99 frac 4 33 end aligned inthanxngediywknsamarthphisucnidwa thsniymsathimichwngsayaw n hlk casamarthekhiyninrupkhxngessswnthimiessepnchudtwelkhthisa aelaswnepn 10n 1 xnukrmkalng aekikh caksutr 1 1 x 1 x x 2 x 3 x 4 x n displaystyle frac 1 1 x 1 x x 2 x 3 x 4 x n emux x lt 1 displaystyle x lt 1 dd samarthnaipphisucnxnukrmxun idodyaekhlkhuls echn emuxnasutrniiphaxnuphnthsa caid 1 1 x 2 1 2 x 3 x 2 4 x 3 5 x 4 n x n 1 displaystyle frac 1 1 x 2 1 2x 3x 2 4x 3 5x 4 nx n 1 emux x lt 1 displaystyle x lt 1 dd 1 1 x 3 2 6 x 12 x 2 20 x 3 30 x 4 n n 1 x n 2 displaystyle frac 1 1 x 3 2 6x 12x 2 20x 3 30x 4 n n 1 x n 2 emux x lt 1 displaystyle x lt 1 dd 1 1 x 4 6 24 x 60 x 2 120 x 3 210 x 4 n n 1 n 2 x n 3 displaystyle frac 1 1 x 4 6 24x 60x 2 120x 3 210x 4 n n 1 n 2 x n 3 emux x lt 1 displaystyle x lt 1 dd epnechnnieruxyip bthkhwamekiywkbkhnitsastrniyngepnokhrng khunsamarthchwywikiphiediyidodyephimkhxmul duephimthi sthaniyxy khnitsastrekhathungcak https th wikipedia org w index php title xnukrmerkhakhnit amp oldid 8265114, wikipedia, วิกิ หนังสือ, หนังสือ, ห้องสมุด,