infi2HW05 - fl fl fl fl fl < ε . n → ∞...

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5 'qn milibxz oeilb - 104281 - 2 itpi` .mixdva 12 : 00 dry cr -- 2008 x`exatl 21 :dybd jix`z A 4 lcebn xiip lr yibdl yi .qxewd ly mi`zd cg`l ec`n` oiipaa qt` dnewa dybdd :zxekfz sca jxev oi` .oeilib xtqne ,f"z ,zeny xexiaa oiivl `p .wcedn yibdl `p .mixqlw e` zeiwy `ll .xry 1 libxz :ze`ad zeivwpetd z` onqp . [ a,b ] rhwa zeiliaxbhpi` zeivwpet g ( x ) - e f ( x ) eidi F ( x ) = Z x a f ( t ) dt, G ( x ) = Z x a g ( t ) dt :miwlga divxbhpi` zgqep z` gikep df libxza Z b a F ( x ) g ( x ) dx = F ( x ) G ( x ) b a - Z b a f ( x ) G ( x ) dx . [ a,b ] rhwa y"na zetivx G ( x ) - e F ( x ) ik egiked .` .miniiw dgqepa milxbhpi`d lk dnl exiaqd .a :`gqepd z` egiked . [ a,b ] rhwd ly dwelg a = x 0 < x 1 < ··· < x n - 1 < x n = b idz .b F ( x ) G ( x ) b a = n X k =1 F ( x k ) Z x k x k - 1 g ( t ) dt + n X k =1 G ( x k - 1 ) Z x k x k - 1 f ( t ) dt m`y jk δ > 0 zniiw ε > 0 lkl :d`ad dprhd z` gikedl zpn lr y"na zetivxa eynzyd .c :if` | x 1 - x 2 | < δ - e a < x 1 < x 2 < b F ( x 1 ) Z x 2 x 1 g ( t ) dt - Z x 2 x 1 F ( t ) g ( t ) dt < ε Z x 2 x 1 | g ( t ) | dt dwelg `id P m`y jk δ > 0 zniiwy egikede mcew sirqa eynzyd . ε > 0 idi .d if` λ ( P ) < δ dwelg xhnxt zlra [ a,b ] ly a = x 0 < x 1 < ··· < x n - 1 < x n = b n X k =1 F ( x k ) Z x k x k - 1 g ( t ) dt - Z b a F ( x ) g ( x ) dx < ε mbe n X k =1 G ( x k - 1 ) Z x k x k - 1 f ( t ) dt - Z b a G ( x ) f ( x ) dx
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Unformatted text preview: fl fl fl fl fl &lt; ε . n → ∞ xy`k λ ( P n ) → y jk [ a,b ] rhwd ly P n zewelg zxcq epz .e .mincewd mitirqd t&quot;r miwlga divxbhpi` zgqep z` egiked .f 1 2 libxz .mi`ad milxbhpi`d ly zeqpkzd ewcia R 1 e x-1 x dx (`) R 1 dx 1-cos x (a) R 1 dx √ 1-x 4 (b) R ∞ 1 arctan x x dx (c) R ∞ 1 dx 1+[ x ] 2 (d) R ∞ 1 x 6 e x dx (e) R ∞-∞ e-x 2 dx (f) 3 libxz . β- e α mixhnxtd t&quot;r miqpkzn milxbhpi`d izn e`vn R π 2-π 2 dx (cos x ) α (`) R ∞ log(1+ x 2 ) x α (a) R ∞ 1 dx x α (log x ) β dx (b) 4 libxz ?qpkzn R ∞ 1 sin( x α ) dx lxbhpi`d α xhnxtd ly mikxr eli` xear 5 libxz :egiked .miqpkzn R ∞ f ( x ) dx- e R ∞ f ( x ) dx oke dtivx zxfbp zlra f ( x ) . lim x →∞ f ( x ) = 0 2...
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infi2HW05 - fl fl fl fl fl &amp;amp;lt; ε . n → ∞...

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