Calculus II Homework: The Integral Test and Estimation of Sums Page 1

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1 Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge Questios Emple (The p series) Get upper d lower bouds o the sum for the p series i= /ip with p = 2 if the th prtil sum is used to estimte the sum. =. = +. Emple Is the followig series diverget or coverget? Emple Is the followig series diverget or coverget? =. Note: this is oe of the more comple problems tht c rise tht uses the itegrl test. Solutios Emple (The p series) Get upper d lower bouds o the sum for the p series i= /ip with p = 2 if the th prtil sum is used to estimte the sum. Sice the p series for p = 2 c be show to coverge usig the itegrl test (see Emple 2), we wt to use the result tht s + f() d s s + + f() d which for the th prtil sum is writte s s + f() d s s + 5 f() d () where f(i) = /i 2.

2 Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge 2 First, we eed the th prtil sum: s = i= i 2 = = 205 Now we eed our itegrl estimtes for the remider. Let s work with geerl lower limit, sice tht is the oly thig tht chges i the two itegrls we must perform. 2 d b 2 d b b b ( ) b b ( b b ) = 5 5 Substitutig ito Eq. () bove, we hve s or.62 s.67 =. The itegrl test requires tht we work with f(), where ) f() =, d o the itervl [, ), f() is: ) cotiuous, ) decresig. Here, f() =, which is cotiuous, positive, d decresig o the itervl [, ). So the itegrl test c be used. d

3 Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge d t = lim ( t = (0 ) = ) Sice the itegrl coverges, the series = coverges by the itegrl test. Note tht! = = +. The itegrl test requires tht we work with f(), where ) f() =, d o the itervl [, ), f() is: ) cotiuous, ) decresig. Here, f() = ( + ), which is cotiuous, positive, d decresig o the itervl [, ). So the itegrl test c be used. ( + ) d + d Substitutio: = t+ lim du u = lim (l u )t+ u = + whe =, u = du = d whe = t, u = t + = lim (l t + l ) =, diverges, sice l s. Sice the itegrl diverges, the series = + Emple Is the followig series diverget or coverget? diverges by the itegrl test

4 Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge The series c be rewritte s =. Use the itegrl test to determie whether the series is coverget or diverget. The itegrl test requires tht we work with f(), where ) f() =, d o the itervl [, ), f() is: ) cotiuous, ) decresig. Here, f() =, which is cotiuous, positive, d decresig o the itervl [, ). So the itegrl test c be used. d d t 2 2 = 2 lim ( t 2 ) = 2 (0 ) = 2 Sice the itegrl coverges, the series = coverges by the itegrl test. Note tht 2! = Emple Is the followig series diverget or coverget? =. Note: this is oe of the more comple problems tht c rise tht uses the itegrl test. Use the itegrl test to determie whether the series is coverget or diverget. The itegrl test requires tht we work with f(), where ) f() =, d o the itervl [, ), f() is: ) cotiuous, ) decresig. Here, f() = l, which is cotiuous d positive o the itervl [, ). But is it decresig o this itervl? It is ot obvious, sice both the umertor d deomitor re icresig fuctios of. However, if fuctio f() is decresig, the it must be true tht f () < 0. Let s tke the derivtive of f() d see wht we c ler. d d f() = d l d = l 2

5 Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge 5 For this to be less th zero, we require l < 0 > e. This will certily be true if >, sice e We c therefore pply the itegrl test to the series work o the itervl [, ). =. Note tht we strt t = d ot =, sice we must l d l Substitutio: l t l d u du = 2 lim l t u2 = 2 lim l ( l 2 t l 2 ) u = l whe =, u = l du = d whe = t, u = l t =, diverges, sice l 2 t s t. Sice the itegrl diverges, the series = diverges by the itegrl test. Therefore, the series = diverges.