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Nadichedevudu Chapters
69. ª«sW¸R…V c ú‡Áx¤¦¦¦ø
ALigýRiª«sVWÌÁLi : ÇÁgRiµæR…VLRiV $$$ ¿RÁLiúµR…ZaP[ÅÁlLi[LiúµR… xqsLRixqs*¼½.
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C aý][NRPª«sVVƒ«sNRPV @LóRi®ªs[Vª«sVƒ«sc
""G xqsLiÅÁùQ\®ƒsƒ«sƒ«sV aRPWƒ«sùª«sVV»][ gRiVßÓáLiÀÁƒ«s aRPWƒ«sùª«sVV @gRiVƒ«sV. aRPWƒ«sùª«sVVƒ«sV G xqsLiÅÁù»][ ˳ØgjiLiÀÁƒ«sƒ«sV aRPWƒ«sùª«sVV ª«s¿RÁV胫sV. NS¬s G xqsLiÅÁùQ\®ƒsƒ«sƒ«sV aRPWƒ«sùª«sVV»][ ˳ØgjiLiÀÁƒ«s¿][ ª«s¿RÁV胫sµj… "ÅÁ¥¦¦¦LRiª«sVV' (ÛÍÁ[NRP @ƒ«sLi»R½ª«sVV) @ƒ«s‡Á²R…Vƒ«sV. C ÅÁ¥¦¦¦LRiª«sVVƒ«sVLi²T… G®µ…[¬s xmsLji„sV»R½xqsLiÅÁùƒ«sV ¼d½zqs®ªs[zqsƒ«sƒ«sV, NRPÖÁzmsƒ«sƒ«sV, µy¬s „sÌÁVª«s ª«sWLRiµR…V. C ÅÁ¥¦¦¦LRiª«sVV @ƒ«sLi»R½ª«sVV, @µj…*¼d½¸R…Vª«sVV @gRiV xmsLRiú‡Áx¤¦¦¦øª«sVV ª«sLiÉÓÁµj…. ú‡Áx¤¦¦¦øª«sVV NRPW²R… ˳ÏÁW»R½xqsXztísQª«sÌÁƒ«s gS¬s, ÌÁ¸R…Vª«sVVª«sÌÁƒ«s gS¬s ª«sWLRiVö ¿ÁLiµR…µR…V.''
C \|ms aý][NRPª«sVVÍÜ[ ˳ØxqsäLS¿yLRiVù²R…V gRißÓá»R½aSryòQûƒ«sVryLRiª«sVV @\®µ…Q*»R½ª«sVVƒ«sV ¬sLRiWzmsLi¿Áƒ«sV. G xqsLiÅÁùQ\®ƒsƒ«sƒ«sV aRPWƒ«sùª«sVV»][ ˳ØgjiLi¿RÁgS ª«s¿RÁV胫sµj… ÅÁ¥¦¦¦LRiª«sV¬s¸R…VV, µk…¬sNTP G xmsLji„sV»R½xqsLiÅÁùƒ«sV NRPÖÁzmsƒ«sƒ«sV µy¬s „sÌÁVª«s |msLRigRiµR…¬s¸R…VV, ¼d½zqs®ªs[zqsƒ«s¿][ µy¬s „sÌÁVª«s »R½gæRiµR…¬s¸R…VV ˳ØxqsäLRiV²R…V ¿Á|msöƒ«sV. @ƒ«sLi»R½ª«sVVƒ«sNRPV @ƒ«sLi»R½ª«sVV NRPÖÁzmsƒ«sƒ«sV @ƒ«sLi»R½®ªs[V ª«s¿RÁV胫sV.
@: 1+2+3+4+....@ƒ«sLi»R½ª«sVV ª«sLRiNRPV NRPW²T…ƒ«s, „sÌÁVª«s @ƒ«sLi»R½ª«sVgRiVƒ«sV.
A: 1+1+1+1+1+.... @ƒ«sLi»R½ª«sVV ª«sLRiNRPV NRPW²T…ƒ«s, „sÌÁVª«s @ƒ«sLi»R½ ª«sVgRiVƒ«sV.
@+ A: 1+2+3+4+5+.....@ƒ«sLi»R½ª«sVV ª«sLRiNRPV NRPW²T…ƒ«s, „sÌÁVª«s @ƒ«sLi»R½ ª«sVgRiVƒ«sV.
C „sµ³R…LigS úxmsÌÁ¸R…Vª«sVVÍÜ[ xmsLji„sV»R½\®ªsVƒ«s ÛÍÁ[NRP @xmsLji„sV»R½\®ªsVƒ«s ª«sxqsVòª«soÌÁV ª«sÀÁè¿Á[Ljiƒ«sƒ«sV, xmsLRiú‡Áx¤¦¦¦øª«sVV FsÉíÓÁ ª«sWLRiVöÛÍÁ[NRP¹¸…[V DLi²R…Vƒ«sV. NRPƒ«sVNRP ú‡Áx¤¦¦¦øª«sVVƒ«sLiµR…V úxms®ªs[bPLi¿RÁV »R½*LixmsµR…ªy¿RÁùª«sVgRiV úxmsxmsLi¿RÁª«sVV ú‡Áx¤¦¦¦ø®ªs[V ¸R…VgRiVƒ«sV. GÌÁ¸R…Vƒ«s c úxmsxmsLi¿RÁª«sVV ª«sÌÁƒ«s ú‡Áx¤¦¦¦øª«sVV ¹¸…VVNRPä ª«sVWÌÁxqs*LRiWxmsª«sVV ƒ«sLiµR…V »R½LRiVgRiVµR…ÌÁgS¬s, |msLRiVgRiVµR…ÌÁ gS¬s ÛÍÁ[µR…V. ®µ…[¬sª«sÌÁƒ«s ª«sVWÌÁª«sxqsVòª«so ¹¸…VVNRPä „sÌÁVª«s ÛÍÁ[NRP xmsLjiª«sWß᪫sVVÍÜ[ |msLRiVgRiVµR…ÌÁ ÛÍÁ[NRP »R½LRiVgRiVµR…ÌÁ DLi²R…µ][, @µj… aRPWƒ«sùª«sVVgS¬s, ª«sW¸R…V (@¼½xqs*ÌÁöQ\®ªsVƒ«s „sÌÁVª«s) gS¬s @LiVV DLi²R…ª«sÛÍÁƒ«sV. ˳ØxqsäLRiV²R…V ¿Ázmsöƒ«s aRPWƒ«sùLi ËݵôðR…VÌÁ¿Á[ @LigkiNRPLjiLixms‡Á²T…ƒ«s aRPWƒ«sùLi NSµR…V. ˳ØxqsäLRiV¬s aRPWƒ«sù®ªs[V @\®µ…Q*»R½VÌÁ ª«sW¸R…V.
ËݵôR…VÌÁ aRPWƒ«sùª«sVV aRPaRP„suyß᪫sVV ÛÍÁ[NRP ª«sLiµ³yùxmsoú»R½V¬s ª«sLiÉÓÁµj…. @ÉíÓÁ aRPWƒ«sùª«sVVƒ«sNRPV gRißÓá»R½ª«sVVÍÜ[ úxms®ªs[aRPª«sVVÛÍÁ[µR…¬s gRißÓá»R½aSxqsòQûÇìÁÙÌÁ @Õ³ÁúFy¸R…Vª«sVV.
˳ØxqsäLRiV¬s aRPWƒ«sùª«sVƒ«sgS 1/n (n"w). B¿RÁèÈÁ 1/n NRPV ª«sVVLiµR…V Lt (@ª«sµ³j…¬s xqsWÀÁLi¿Á[ gRiVLRiVò) @ƒ«sV gRiVLRiVò DLi²R…µR…V. GÌÁ¸R…Vƒ«s c Lt (n")1/n @ƒ«sgS aRPWƒ«sùª«sVV (4c 4 = 0 ª«sLiÉÓÁµj…) @gRiVƒ«sV. NS¬s 1/n (n") @˳ت«sLRiWxmsª«sVV NSµR…V. @µj… ˳ت«sLRiWxmsª«sVV. µy¬s „sÌÁVª«s ª«sWú»R½ª«sVV @¼½xqs*ÌÁöª«sVV. BÉíÓÁµj…¹¸…[V ª«sW¸R…V NRPW²y. NRPƒ«sVNRP ˳ØxqsäLRiV¬s "ÅÁ¥¦¦¦LRiª«sVV' 1/n (n"0). NS¬s, 1x n (n"w) NS¬s @gRiVƒ«sV. NRPƒ«sVNRP ú‡Áx¤¦¦¦øª«sVV INRP @ª«sµ³j… ª«sWú»R½®ªs[V.
ZNP[ƒ¯[xms¬sxtsQ»R½Vò xmsLRiú‡Áx¤¦¦¦øª«sVVƒ«sV gRiWLjiè BÈýÁV ¿ÁxmsöVƒ«sV.
""ƒyx¤¦¦¦Li ª«sV®ƒs[ù xqsV®ªs[®µ…[¼½ ƒ¯[ ƒ«s®ªs[®µ…[¼½ ®ªs[µR… ¿RÁe
¹¸…Wƒ«s xqsò®µ…[*µR… »R½®µ…[*µR… ƒ¯[ ƒ«s®ªs[®µ…[¼½ ®ªs[µR… ¿RÁee''
""ƒyNRPV ¾»½ÖÁ¸R…VVƒ«s¬s ®ƒs[ƒ«sV ˳؄sLi¿RÁVÈÁ ÛÍÁ[µR…V. ƒyNRPV ¾»½ÖÁ¸R…VµR…¬s NRPW²y ƒyNRPV ¾»½ÖÁ¸R…VµR…V. ª«sVƒ«sÍÜ[ Fsª«s¬sNTP ¾»½ÖÁ¸R…VVƒ¯[, @»R½¬sNTP ¾»½ÖÁ¸R…VVƒ«sV. Fsª«s¬sNTP ¾»½ÖÁ¸R…Vµ][, @»R½¬sNTP ¾»½ÖÁ¸R…VµR…V.''
""¸R…Vryù ª«sV»R½Li »R½xqsù ª«sV»R½Li, ª«sV»R½Li ¸R…Vxqsù ƒ«s®ªs[µR…xqsM
@„sÇìØ»R½Li „sÇ؃«s»yLi „sÇìØ»R½ ª«sV„sÇ؃«s»yª±sVee''
""xmsLRiú‡Áx¤¦¦¦ø ¾»½ÖÁ¸R…VVÈÁNRPV aRPNRPùª«sVVNS¬sµj… @¬s @LóRiª«sVV ¿Á[xqsVNRPVƒ«sõªy²R…V µy¬s¬s ¾»½ÌÁVxqsVNRPVƒ«sõªy®²…[ ¸R…VgRiVƒ«sV. ®ƒs[ƒ«sV ú‡Áx¤¦¦¦øƒ«sV ¾»½ÌÁVxqsVN]LiÉÓÁ¬s @¬s ˳؄sLi¿RÁVªy¬sNTP @µj… ¾»½ÖÁ¸R…VµR…V. ú‡Áx¤¦¦¦ø, Çì؃«s„sxtsQ¸R…VLi NSµR…¬s ú‡Áx¤¦¦¦øÇì؃«sVÌÁ @Õ³ÁúFy¸R…Vª«sVV. ú‡Áx¤¦¦¦ø, Çì؃«s„sxtsQ¸R…V®ªs[V @¬s @Çì؃«sVÌÁV »R½ÌÁLi»R½VLRiV.''
gRißÓá»R½aSxqsòQûª«sVVƒ«sNRPV ¿ÁLiµj…ƒ«s INRP µR…XuíyLi»R½ª«sVV µy*LS \|msƒ«s ¿Ázmsöƒ«s úaRPV¼½ ªyNRPùª«sVVÌÁ ˳ت«sª«sVV xqsVÌÁ˳ÏÁª«sVVgS ¾»½ÖÁ¸R…Vƒ«sgRiVƒ«sV. INRP ª«sùQQNTPò @ƒ«sLi»R½ª«sVV ƒyNRPV ¾»½ÖÁ¸R…VVƒ«s¬s ¿ÁxmsöV ƒ«sƒ«sVN]ƒ«sV²R…V. @»R½¬sNTP @µj… ¾»½ÖÁ¸R…VÛÍÁ[ µR…ƒ«sõª«sWÈÁ. @ƒ«sLi»R½ª«sVV ¾»½ÖÁ¸R…VVÈÁNRPV ryµ³R…ùª«sVVgS¬sµj… @¬s @»R½²R…V ¿Ázmsöƒ«s¿][, @»R½¬sNTP @µj… ¾»½ÖÁzqsƒ«sÛÉýÁ[. JLiNSLRiª«sVV xmsLRiú‡Áx¤¦¦¦øNRPV INRP ¿RÁNRPä¬s aRP‡ôÁúxms¼d½NRP. @ÈÁVÌÁ®ƒs[ @ƒ«sLi»R½ª«sVV gRißÓá»R½aSxqsòQûª«sVVƒ«sNRPV ¿ÁLiµj…ƒ«s xmsLRiú‡Áx¤¦¦¦ø úxms¼d½NRP.
˳ØxqsäLRiV²R…V BÈýÁV ¿Á|msöƒ«sV.
G®µ…[¬s xqsLiÅÁù x aRPWƒ«sùª«sVV = aRPWƒ«sùª«sVV
G®µ…[¬s xqsLiÅÁù ÷ aRPWƒ«sùª«sVV = ÅÁ¥¦¦¦LRiª«sVV (@ƒ«sLi»R½ª«sVV)
C lLiLi²R…VªyNRPùª«sVVÌÁV ®ªs[lLi[*LRiVgS NRPƒ«s‡Á²T…ƒ«sƒ«sV, IZNP[ @\®µ…Q*»R½zqsµôðyLi»R½xqs»R½ùª«sVVƒ«sV úxms¼½Fyµj…Li¿RÁV¿RÁVƒ«sõ„s. C lLiLi²R…V gRißÓá»R½xqsWú»R½ª«sVVÌÁƒ«sV @\®µ…Q*»R½xmsLRiª«sVVgS BÈýÁV ªyùÆØù¬sLi¿RÁª«s¿RÁV胫sV.
a / D a = w ÛÍÁ[NRP a = w x D a
@ƒ«sgS @ƒ«sLi»R½ª«sVV (xmsLRiú‡Áx¤¦¦¦ø) x aRPWƒ«sùª«sVV (ÅÁ ÛÍÁ[NRP ª«sW¸R…V) = xmsLjiÀÁè郫sõ úxmsxmsLi¿RÁª«sVV.
"a' @ƒ«sV xmsLjiÀÁè郫sõxqsLiÅÁù „sƒ«sW»R½õQQ\®ªsVƒ«sµj… G„sV¸R…VV NSµR…V. @µj… "ÅÁ¥¦¦¦LRiª«sVV' ¹¸…VVNRPä @ƒ«sLi»R½\®ªsVƒ«s „sÌÁVª«sÍÜ[ @Li»R½LS÷égRiª«sVV ª«sWú»R½®ªs[V. @ÈÁVÌÁ®ƒs[ ú‡Áx¤¦¦¦øª«sVV ƒ«sVLi²T… ª«sW¸R…WxqsLi‡ÁLiµ³R…ª«sVV ª«sÌÁƒ«s A„sLRi÷é„sLiÀÁƒ«s xmsLjiÀÁè郫sõQúxmsxmsLi¿RÁª«sVV ¸R…Vµ³yLóRiª«sVVgS xqs»R½ùLRiWxmsª«sVgRiV xmsLRiú‡Áx¤¦¦¦øª«sVV NRPLiÛÉÁ Õ³Áƒ«sõª«sVVNSµR…V. NRPƒ«sVNRP®ƒs[ ˳ØxqsäLRiV²R…V »R½ƒ«s ÅÁ¥¦¦¦LRiª«sVVƒ«sV @ƒ«sLi»R½LRiª«sVV @¿RÁVù»R½ª«sVVƒ«sgRiV xmsLRiú‡Áx¤¦¦¦øª«sVV»][ F¡ÛÍÁ胫sV. GÌÁ¸R…Vƒ«s c xmsLRiú‡Áx¤¦¦¦ø ˳ÏÁW»R½xqsXztísQ ª«sÌÁƒ«sgS¬s, @ƒ«sLi»R½Ë³ÏÁW»R½gRiVß᪫sVVÌÁV »R½ƒ«s¸R…VLiµR…V ÖdÁƒ«sª«sVgRiVÈÁ ª«sÌÁƒ«sgS¬s, ª«sWLRiVö¿ÁLiµR…µR…V.
ú‡Áx¤¦¦¦øª«sVV (xqs»`½), "GNRPª«sVV', @¬s úaRPV»R½VÌÁ¸R…VLiµR…V xms®µ…[xms®µ…[ ¿Áxmsö‡Á²R…Vƒ«sV. C "GNRPª«sVV' xqsLiÅÁùÌÁÍÜ[ 2ÍÜ[ xqsgRiª«sVV, 4ÍÜ[ ƒyÌæÁª«s ª«sLi»R½V @LiVVƒ«s "1' NSµR…V. G @ƒ«sLi»R½ª«sVVƒ«sLiµR…V xmsLjiÀÁè郫sõxqsLiÅÁù ÌÁ¬sõ¸R…VV ÌÁ¸R…Vª«sVgRiVƒ¯[ A @ƒ«sLi»R½ª«sVV C GNRPxmsµR…ª«sVVƒ«s NRPLóRiª«sVV. ˳ØxqsäLRiV²R…V "ÖdÁÍت«s¼½'ÍÜ[¬s @Li¼½ª«sVaý][NRPª«sVVƒ«sLiµR…V xmsLRiú‡Áx¤¦¦¦ø¬sÈýÁV ª«sLñjiLi¿Áƒ«sV.
"@uí¢ªyùNRPLRißجs xtsQÉÞ ¿RÁ Õ³ÁxtsQÇØLi ªyù¿RÁxtísQ »yxqs=Liz¤¦¦¦»yM
xtsQÉÞ »R½LS䃱s gRißÓá»y¬s xmsLi¿RÁ ¿RÁ»R½VlLi[*µyƒ«s ´k… ¾»½[ xqsø¸R…VLiiMe
LRi»yõƒyLi ú¼½»R½¸R…VLi µR…*¸R…VLi ¿RÁ ‡ÁV‡ÁV®µ³…[ „dsVª«sWLixqs ¹¸…WLRiLi»R½LRiLi
xqsµR…÷QûQ\|¤¦¦¦øNRP ª«sVgSµ³R… ËÜ[µ³R… ª«sVz¤¦¦¦ª«sW r¡s ryùM NRP„s LS÷éxqsäLRiMee'
ÕdÁÇÁgRißÓá»R½Li ÍÜ[¬s 11ª«s aý][NRPLiÍÜ[ xmsLRiú‡Áx¤¦¦¦ø®ƒs[ "@ƒ«sLi»R½' "@¿RÁVù»R½' @®ƒs[ }msLýRi»][ }msL]䮃sƒ«sV. @LiµR…V¿Á[»R½ ú‡Áx¤¦¦¦øNRPV ¿Ázmsöƒ«s C "GNRP»R½*Li' "1' @®ƒs[ xqsLiÅÁùÍÜ[¬s GNRP»R½*Li NSµR…V. Bµj… gRißÓá»R½aSxqsòQûLiÍÜ[¬s @ƒ«sLi»R½ª«sVV ÛÍÁ[NRP @\®µ…Q*»R½ª«sVV.
gRiª«sV¬sNRP 2/0(ÅÁ) = @ƒ«sLi»R½ª«sVV (ÅÁ¥¦¦¦LRiª«sVV)
@ƒ«sLi»R½ª«sVV x 0 = 2 ÛÍÁ[NRP G®µ…[¬s xqsLiÅÁù.
ú‡Áx¤¦¦¦ø x ª«sW¸R…V = xmsLjiÀÁè郫sõ úxmsxmsLi¿RÁª«sVV.
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Nadichedevudu Chapters
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