MAT107 2.pdf - MAT107 Vr 2017 Innlevering 2 Hgskulen p...

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Høgskulen på Vestlandet Institutt for data- og realfag MAT107 Vår 2017 Innlevering 2 LØSNINGSFORSLAG 1 Finn konvergensradien og konvergensintervallet for potensrekkene: i) X n =1 1 n x n ii) X n =1 3 n · n n 2 + 1 x n Løsning: i) lim n →∞ a n +1 a n = lim n →∞ 1 n + 1 n 1 = lim n →∞ r n n + 1 = 1 Konvergensradien er R = 1 lim n →∞ a n +1 a n = 1 1 = 1 Senteret for konvergensintervallet er c = 0 . Rekken konvergerer absolutt på intervallet (0 - 1 , 0 + 1) = ( - 1 , 1) . Sjekker endepunktene også. For x = - 1 : X n =1 1 n ( - 1) n er en alternerende rekke, konvergerer ved Leibniz testen (alternerende rekke testen). For x = 1 : X n =1 1 n 1 n = X n =1 1 n er en divergent p -rekke. Konvergensintervallet er [ - 1 , 1) . ii) lim n →∞ a n +1 a n = lim n →∞ 3 ( n +1) · ( n + 1) ( n + 1) 2 + 1 n 2 + 1 3 n · n = lim n →∞ 3 · ( n + 1)( n 2 + 1) (( n + 1) 2 + 1) n = 3 Konvergensradien er R = 1 lim n →∞ a n +1 a n = 1 3 Senteret for konvergensintervallet er c = 0 . Rekken konvergerer absolutt på interval- let (0 - 1 3 , 0 + 1 3 ) = ( - 1 3 , 1 3 ) . Sjekker endepunktene også. For x = - 1 3 : X n =1 3 n · n n 2 + 1 ( - 1 3 ) n = X n =1 ( - 1) n · n n 2 + 1 er en alternerende rekke, konverge- rer ved Leibniz testen (alternerende rekke testen). For x = 1 3 : X n =1 3 n · n n 2 + 1 ( 1 3 ) n = X n =1 n n 2 + 1 divergerer (bruk sammenligningstesten med grenseverdi og sammenlign med 1 n ). Konvergensintervallet er [ - 1 3 , 1 3 ) . 1. mars 2017 Side 1 av 7
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Innlevering 2 LØSNINGSFORSLAG 2 a) Finn en potensrekke for funksjonen: e - x 3 - 1 x 3 . Ikke bare skriv de første leddene, få med ledd n . Løsning: Begynner med MacLaurin rekken for e - x : e - x = X n =0 ( - x ) n n ! = X n =0 ( - 1) n n !
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