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المرحلة 1
أستاذ المادة فؤاد حمزة عبد الشريفي
15/03/2017 17:56:04
Infinite Series An infinite series is given by the terms of an infinite sequence, added together. For example, we could take the infinite sequence {2n}={2 ,4 ,6 ,?} Then the corresponding example of an infinite series would be given by all of these terms added together 2+4+6+8+? So we would have ?_(n=1)^??2n=2+4+6+8+? If ??a_n is a positive series,then either ??a_n converges to a positive number, or??a_n diverges to infinity. Theorem: The nth-term test for divergence.If lim?(n??)??a_n ??0 ,or if lim?(n??)??a_n ? not exist then ??a_n diverges. For example the series? ?_(n=1)^??n^2 diverges because lim?(n??)??n^2 ?=??0 ?_(n=1)^??(n+2)/n diverges because lim?(n??)??(n+2)/n?=1?0 ?_(n=1)^??(-1)^(n+1) diverges because lim?(n??)??(-1)^(n+1) ? dose not exist
Geometric Series A geometric series is any series that can be written in the form, ?_(n=1)^???ar^(n-1) ?=a+ar+ar^2+ar^3+?+ar^n+? 1. If |r|<1 then the geometric series converges to the sum s_n=a/(1-r) 2. If |r|?1 then the geometric series diverges to ? . For example the series ?_(n=1)^??( 1 )/2^n =( 1 )/( 2 )+( 1 )/( 4 )+( 1 )/( 8 )+( 1 )/( 16 )+? converges because r=1/( 2 )<1 and the sum is s_n=(1?2)/(1-(1?2) )=1. The series ?_(n=1)^??(3/( 2 ))^n diverges to ? because r=3/( 2 )>1 The p-series If p is a real constant,the series?_(n=1)^??( 1 )/n^p =( 1 )/1^p +( 1 )/2^p +( 1 )/3^p +? 1.converges if p>1 . 2. diverges if p?1 . For example the series ?_(n=1)^??1/n^2 converges because p=2>1 The series ?_(n=1)^??1/?(n )=?_(n=1)^??1/n^(1?2) diverges to ? because p=1/( 2 )<1
Tests for converges of series 1. Integral Test Let the function f(x)=a_n (x) be continuous, positive and decreasing then the series?_(n=1)^??a_n and the integral ?_1^??f(x) dx both converge or both diverge. Example 1: Determine whether the series converges or diverges. 1. ?_(n=1)^??1/(n^2+1) 2. ?_(n=1)^??n/(n^2+1) 3. ?_(n=1)^??1/(n^2-n-3) 1. ? ?_1^??dx/(x^2+1)=tan^(-1)?x ?|_1^?=?/2-?/4=?/4 converge 2.?_1^??xdx/(x^2+1)=? 1/2 ln?|x^2+1| ?|_1^? =? diverge 3. 1/(x^2-x-2)=A/(x-2)+ B/(x+1) ? x=2 ?A=( 1 )/( 3 ) and x=-1 ?B=-( 1 )/( 3 ) ?_1^??dx/(x^2-x-2)=( 1 )/( 3 ) ? ?_1^??(1/(x-2)- 1/(x+1)) dx=( 1 )/( 3 ) ln?|(x-2)/(x+1)| ?|_1^? =? ( 1 )/( 3 ) ln?|(1-( 2 )/x)/(1+( 1 )/x)| ?|_1^?=( 1 )/( 3 ) ln?2 converge 2. Ratio Test Let ?_(n=1)^??a_n be a series with non-negative terms and lim?(n??)??a_(n+1)/a_n ?=L . If 1. L<1 then the series converges 2. L>1 then the series diverges 3. L=1 then this test is inconclusive Example 2: Determine whether the series converges or diverges. 1. ?_(n=1)^??? n?^3/( 3^n ) 2. ?_(n=1)^??n!/2^n 3. ?_(n=1)^??3/(2n+5) 4.?_(n=1)^??5^n/(n 3^n ) 1. lim?(n??)??a_(n+1)/a_n ?=lim?(n??)??? (n+1)?^3/( 3^(n+1) )×( 3^n)/? n?^3 ?=( 1 )/( 3 ) lim?(n??) ((n+1)/n)^3 =( 1 )/( 3 ) lim?(n??) (1+( 1 )/n)^3=( 1 )/( 3 )<1 converges 2. lim?(n??)??a_(n+1)/a_n ?= lim?(n??) (n+1)!/2^(n+1) ×(2^n )/n! =lim?(n??) ((n+1)×n!)/(2^n×2)×(2^n )/n!=lim?(n??) (n+1)/2=?>1 diverges
3. lim?(n??)??a_(n+1)/a_n ?=lim?(n??)??3/(2(n+1)+5)?×( 2n+5 )/( 3 ) =lim?(n??)??(2n+5)/(2n+7)?=1. The test is inconclusive . So we try another test ? ?_1^??3dx/(2x+5)=( 3 )/( 2 ) ln?|2x+5| ?|_1^?=? diverges 4. lim?(n??)??a_(n+1)/a_n ?=lim?(n??)??5^(n+1)/((n+1) 3^(n+1) )×(n 3^n)/5^n ? =lim?(n??)??5n/(3(n+1) )?=( 5 )/( 3 )>1 diverges 3. Root Test Let ?_(n=1)^??a_n be a series with non-negative terms and lim?(n??)??(n&a_n )=L . If 1. L<1 then the series converges 2. L>1 then the series diverges 3. L=1 then this test is inconclusive Example 3: Determine whether the series converges or diverges. 1. ?_(n=1)^??((3n+2)/( 4n+1))^n 2. ?_(n=1)^??(1/( n ))^n 3. ?_(n=1)^??2^n/n^2 4. ?_(n=1)^??((n-1)/( n ))^n 1. lim?(n??)??(n&a_n )=lim?(n??)??(3n+2)/( 4n+1)?=( 3 )/( 4 )<1 converges
2. lim?(n??)??(n&a_n )= lim?(n??) 1/( n )=0<1 converges 3. lim?(n??)??(n&a_n )=lim?(n??)??2/?(n&n^2 )?=2/(lim?(n??) ?(n&n))=( 2 )/( 1 )=2 >1 diverges 4. lim?(n??)??(n&a_n )=lim?(n??) (n-1)/( n )=1.The test is inconclusive . So we try another test lim?(n??) ((n-1)/( n ))^n=lim?(n??) (1+(-1)/( n ))^n=e^(-1)?0 diverges Find the sum of the following series: 1. ?_(n=1)^??( 2^n )/? 5?^n 2.?_(n=1)^??( 5^n )/? 3?^2n 3.?_(n=1)^??( 4 )/? 3?^(n+1) Determine whether the series converges or diverges. Give reasons for your answers. 4. ?_(n=1)^??(n/( 2n-1 ))^n 5. ?_(n=1)^??( 1 )/(? n?^2 ?n) 6. ?_(n=1)^??(n-1)!/( n !) 7. ?_(n=1)^??( ?11?^n )/? 3?^2n 8. ?_(n=1)^???ne^(-n^2 ) ? 9.?_(n=1)^??? e^(-?n)/?n? 10. ?_(n=1)^??((3n+4)/( 2n ))^(-n) 11. ?_(n=1)^??1/( (3n+1)^3 ) 12. ?_(n=1)^????(n+3)!/(2^n n!)
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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