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# 07Wb - (15!mkicrv!`ad cenra jynd 1 5 dl`y xnyn `ed f` D...

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30 . 5 . 2008 'a cren ,g"qyz sxeg 2 itpi` ipinipa a`ei :dxend myd aezkle cala legk e` xegy hra ynzydl `p .xfr xneg meya ynzydl oi` .zery yely dpigad jyn .mkicrv z` ahid ehxte ewnp .cenr lk y`xa ...100 "wx" `ed ilniqknd oeivd j` ,103 `ed llekd cewipd .sirq lka riten cewipd !dglvda . 1 dl`y .qpkzn a n xehdy jk zcxei zipehepen ziaeig dxcq { a n } idz (10%) ( i ) . lim n →∞ na n = 0 ik egiked lxbhpi`dy jk [0 , ) lr zcxei zipehepen ziaeig divwpet f idz (5%) ( ii ) . lim x →∞ xf ( x ) = 0 ik egiked .qpkzn R 0 f ( x ) dx llkend llkend lxbhpi`dy jk (0 , 1] lr zcxei zipehepen ziaeig divwpet f idz (5%) ( iii ) . lim x 0 + xf ( x ) = 0 ik egiked .qpkzn R 1 0 f ( x ) dx . 2 dl`y dcina f n f m`e , [0 , 1] rhwa onix zeiliaxbhpi` f n m` ik egiked (15%) ( i ) miiwzne onix ziliaxbhpi` f mb f` ,rhwa deey . lim n →∞ Z 1 0 f n Z 1 0 f f n ( x ) 0 -y jk [0 , 1] rhwa zetivx f n zeivwpet zxcql `nbec e`iad (5%) ( ii ) . R 1 0 f n 6→ 0 j` ,rhwa x lkl . 3 dl`y . R π/ 2 0 R π/ 2 y sin x x + y dx · dy lxbhpi`d z` eayg (15%) . 4 dl`y z` wiecna ewnpe) f ( x ) = X n =2 ( n - 1) x n +1 divwpetl zyxetn dgqep epz

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Unformatted text preview: (15%) .(!mkicrv !`ad cenra jynd 1 . 5 dl`y xnyn `ed f` D megza l`ivphet F ixehwee dcyl yi m` ik egiked (9%) ( i ) . D-a dliqnd z` γ ( t )-a onqpe ,ixehwe dcy F ( x,y ) = ( xy 2 ,x 2 y + y ) idi (9%) ( ii ) . R γ F z` eayg . ≤ t ≤ π 2 xear γ ( t ) = (2cos t, sin 2 t ) .dxvwa ewnpe zeicbp ze`nbec e`iad .zepekp opi` ze`ad zeprhd . 6 dl`y llkend lxbhpi`dy jk [0 , ∞ ) lr zilily-i`e dtivx divwpet f idz (5%) ( i ) .qpkzn ∑ ∞ n =1 f ( n ) xehd mb f` .qpkzn R ∞ f zeiwlgd zexfbpdy jk xeyina P dcewpd ly V daiaqa zxcben f idz (5%) ( ii ) . P-a dtivx f f` . V daiaqd lka zeniiw f y-e f x-a dxifb F f` . F ( x ) = R x a f onqpe , [ a,b ] rhwa ziliaxbhpi` f idz (5%) ( iii ) . ( a,b ) 2...
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07Wb - (15!mkicrv!`ad cenra jynd 1 5 dl`y xnyn `ed f` D...

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