10. 3 The Integral and Comparison Test, Estimating Sums

Transcription

1 0. The Itegrl d Comriso Test, Estimtig Sums I geerl, it is hrd to fid the ect sum of series. We were le to ccomlish this for geometric series d for telescoig series sice i ech of those cses we could fid simle formul for the rtil sum s. But usully it is ot esy to comute lim s. I the sectio d the et, we develo tests tht ele us to determie whether series is coverget or diverget without elicitly fidig its sum. I this sectio, we just del with series re icresig. By the Mootoic Sequece Theorem, with ositive terms 0, so the rtil sums is coverget if the series is ouded. Result (Itegrl Test): Suose f is cotiuous, ositive, decresig fuctio o [, ) d let f ( ). The th. If. If f ( ) d is coverget, the f ( ) d is diverget, the is coverget is diverget Proof: Let s i f () f () f ( ). i f ( k ) f ( ) f ( k) for k k sice f ( ) is decresig. k k k f ( k ) f ( k ) d f ( ) d f ( k) d f ( k) f () f () f ( ) f ( ) d f () f () f () f ( ) s f () s f ( ) d s f ( ) s f ( ) d s k k k f ( ) d f ( ) d lim s f ( ) d

2 Thus, if f ( ) d is coverget (diverget), the is coverget (diverget). Remrk: The ove result c esily eted to the followig. Result (Itegrl Test: other versio). Suose f is cotiuous, ositive, decresig fuctio o [ k, ) d let f ( ). The. If f ( ) d is coverget, the k. If f ( ) d is diverget, the k is coverget is diverget Result (-series): The -series Proof. is coverget if, d. Cse : 0. Whe 0, test, the series is diverget. lim 0 d whe 0, lim 0, so y the divergece Cse : 0. Whe 0, the the fuctio [, ). From the result of sectio 8.9, we hve d is coverget if, is diverget if. f( ) is decresig, cotiuous d ositive o Thus, we hve is coverget whe, d diverget whe y the Itegrl Test. Result (Comriso Test): Suose tht d re series with ositive terms. If 0. If. If is coverget, the is diverget, the is coverget. is diverget. Result (Limit Comriso Test): Suose tht d re series with ositive terms. If

3 lim c 0 ( c is fiite umer) the either oth series d re coverget or oth diverget. Emle : Determie the followig series re coverget or diverget... c. d. e. f. l l g. si Solutio:. Method : Sice, d ( -series with ) is coverget, thus, is coverget (y Comriso Test). Method : Sice lim lim 0, d is coverget so is coverget (y Limit Comriso Test).

4 . Sice lim lim lim lim 0, d is diverget, so is diverget (y Limit Comriso Test). Plese idetify the mistkes I mde the followig two rocedures (oth re wrog).. Sice, d is coverget (y Limit Comriso Test).. Sice, d diverget (y Limit Comriso Test). is coverget, so is diverget, so is c. Sice l for, d is diverget, so l d. Use The Itegrl Test. Let f( ) (l ). Cosider M is diverget. M d lim d (Let u l, the du d) (l ) (l ) l M l M = lim du lim lim M l u M u M l l l M l Thus, (l ) d is coverget, so l is coverget. Plese idetify the mistkes I mde the followig rocedure (It s wrog). Sice for, d l is diverget, so l is diverget. Questio: Plese determie l f( ). (Aswer: the series is diverget.) (l ) is coverget or diverget. (Use The Itegrl Test, let

5 e. Method : Sice, d = (geometric series with r <) is coverget, thus, is coverget (y Comriso Test). Method : Sice lim lim 0 (sice lim lim lim 0), (l ) d is coverget so is coverget (y Limit Comriso Test). f. Sice lim lim lim lim lim 0 (sice lim lim lim lim 0) l is coverget so is coverget (y Limit Comriso Test). Plese idetify the mistkes I mde the followig rocedure (It s wrog). Sice, d is coverget so is coverget. g. Method. Sice si (use si for 0), d si is coverget. si si Method. Sice lim lim 0, d si is coverget. is coverget, so is coverget, so

6 Emle : Remider Estimte for the Itegrl Test. If R s s, the k is coverget y the Itegrl Test d f ( ) d R s s f ( ) d s f ( ) d s s f ( ) d. Aroimte the sum of the series y usig the sum of the first 0 terms. Estimte the error ivolved i this roimtio.. How my terms re required to esure tht the sum is ccurte to withi 0.000? Solutio: I oth rts () d (), we eed to kow f ( ) d with f( ), or d. T d lim d lim lim T T T T T. 0 s Accordig to the remider estimte, we hve R0 s s0 f ( ) d (0) So the error is t most Accurcy to withi mes tht we hve to fid vlue of such tht R Sice R f ( ) d, we wt Solvig this, we hve (0.000) So, we eed terms to esure ccurcy to withi