The Integral and Comparison Tests

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1 Sectio 8.3 The Itegral ad Compariso Tests The Itegral ad Compariso Tests The Itegral Test 00 Kiryl Tsishchaka THE INTEGRAL TEST: Suppose f is a cotiuous, positive, decreasig fuctio o [, ) ad let a = f(). The the series a is coverget if ad oly if the improper itegral = f()d is coverget. I other words: If If f()d is coverget, the f()d is diverget, the a is coverget. = a is diverget. = REMARK: Do t use the Itegral Test to evaluate series, because i geeral a f()d = EXAMPLES:. is diverget, because f() = is cotiuous, positive, decreasig ad = diverget by the p-test for improper itegrals, sice p =.. = is diverget, because f() = is cotiuous, positive, decreasig ad / / is diverget by the p-test for improper itegrals, sice p = /. 3. = d is /d is coverget, because f() = is cotiuous, positive, decreasig ad d is coverget by the p-test for improper itegrals, sice p = >. REMARK : Whe we use the Itegral Test it is ot ecessary to start the series or the itegral at =. For istace, i testig the series we use + + d =5 REMARK : It is ot ecessary that f be always decreasig. It has to be ultimately decreasig, that is, decreasig for larger tha some umber N. 5 EXAMPLE: Determie whether the series = l coverges or diverges.

2 Sectio 8.3 The Itegral ad Compariso Tests 00 Kiryl Tsishchaka EXAMPLE: Determie whether the series = l coverges or diverges. Solutio: The fuctio f() = l is positive ad cotiuous for >. However, lookig at the graph of this fuctio we coclude that f is ot decreasig. At the same time oe ca show that this fuctio is ultimately decreasig. I fact, ( ) l f () = = (l ) l = l = l Note that l < 0 for all sufficietly large which meas that f () < 0 ad therefore f is ultimately decreasig. So, we ca apply the Itegral Test: l t d t l d t (l) ] t (lt) t = Sice this itegral diverges, the series = l also diverges. EXAMPLE: Determie whether the series l coverges or diverges. Solutio: The fuctio f() = is cotiuous, positive ad decreasig o [, ), therefore l we ca apply the Itegral Test: l t d t d l l(l t )]t [l(l t) l(l)] = t Sice this itegral diverges, the series l also diverges. EXAMPLE: Determie whether the series l coverges or diverges.

3 Sectio 8.3 The Itegral ad Compariso Tests 00 Kiryl Tsishchaka EXAMPLE: Determie whether the series l coverges or diverges. Solutio: The fuctio f() = l is cotiuous, positive ad decreasig o [, ), therefore we ca apply the Itegral Test: l t d t Sice this itegral coverges, the series metioed before, l l. THEOREM (p-test): The p-series [ l d ] t [ t l t l t + ] = l l = l also coverges. We also ote that, as it was is coverget if p > ad diverget if p. p Proof: We distiguish three cases: Case I: If p < 0, the lim =, therefore diverges by the Divergece Test. p p Case II: If p = 0, the lim p Divergece Test. = 0 =, therefore = diverges by the p Case III: If p > 0, the the fuctio f() = is cotiuous, positive ad decreasig o p [, ), therefore we ca apply the Itegral Test by which is coverget if ad oly if p the improper itegral pd is coverget. But if p by the p-test for improper itegrals. = pd is coverget if p > ad diverget REMARK: As before, whe we use the p-test it is ot ecessary to start the series at =. EXAMPLE: Determie whether the followig series coverge or diverge: = Solutio: The series = = = +ǫ, ǫ > 0 diverges by the p-test for series, sice p =. The series coverges by the p-test for series, sice p = >. The series for series, sice p = + ǫ >. EXAMPLE: Determie whether the series = + 3 = coverges or diverges. = coverges by the p-test +ǫ

4 Sectio 8.3 The Itegral ad Compariso Tests 00 Kiryl Tsishchaka EXAMPLE: Determie whether the series = + coverges or diverges. REMARK: Note that we CAN T apply the p-test directly! Solutio : The fuctio f() = is cotiuous, positive ad decreasig o [, ), therefore we ca apply the Itegral + Test: + t d t Sice this itegral diverges, the series Solutio: We have d + = = l( + t )]t [l(t + ) l] = t + also diverges. + = Sice diverges by the p-test with p =, also diverges, sice + = covergece or divergece is uaffected by deletig a fiite umber of terms. The Compariso Tests THE COMPARISON TEST: Suppose that a ad b are series with positive terms. If b is coverget ad a b for all, the a is also coverget If b is diverget ad a b for all, the a is also diverget EXAMPLE: Use the Compariso Test to determie whether the followig series coverge or diverge. l 3 = Solutio: Sice l > for > e ad = l also diverges. Similarly, sice 3 > = 3 ad p = /3, 3 also diverges. = diverges by the p-test with p =, 3 diverges by the p-test with EXAMPLE: Use the Compariso Test to determie whether the followig series coverge or diverge:

5 Sectio 8.3 The Itegral ad Compariso Tests 00 Kiryl Tsishchaka EXAMPLE: Use the Compariso Test to determie whether the followig series coverge or diverge: Solutio: Sice = < 3 ad = 3 coverges by the p-test with p = 3 = also coverges. Sice Sice + < = + ad = also coverges. coverges by the p-test with p = < for ad = coverges by the p-test with p = also coverges. Aother way to show it is to rewrite Sice = + < = + ( + ) = ad as = ( ad therefore + + = = + coverges by the p-test with p = ) also coverges. 5

6 Sectio 8.3 The Itegral ad Compariso Tests 00 Kiryl Tsishchaka THE LIMIT COMPARISON TEST: Suppose that a ad b are series with positive terms. If a lim = c b where c is a fiite umber ad c > 0, the either both series coverge or both diverge. EXAMPLE: Use the Limit Compariso Test to determie whether diverges. Solutio: Put a =, b =. The a c b Sice c = ad coverges. = coverges or = 0 = coverges by the p-test with p =, also EXAMPLE: Use the Limit Compariso Test to determie whether the followig series coverge or diverge: (d) / = = 6

7 Sectio 8.3 The Itegral ad Compariso Tests 00 Kiryl Tsishchaka EXAMPLE: Use the Limit Compariso Test to determie whether the followig series coverge or diverge: (d) / = Solutio: Put a = +, b =. The a c b Sice c = ad diverges. Put a = = a c b Sice c = 4 ad coverges. + = + + diverges by the p-test with p = /, 4, b =. The = + = = + also 4 = 4 coverges by the p-test with p =, 4 also Put a = , b = 3. The a c b Sice c = 4 ad = also coverges. (d) Put a = +/, b =. The = coverges by the p-test with p = 3, 3 a c b +/ / / = / To evaluate lim / we apply L Hospital s Rule. To this ed we first put y = /. The l y = l( / ) = l We have l lim (l) / = 0 Therefore lim / = e 0 =. Sice c = ad diverges by the p-test with p =, it = follows that also diverges. +/ = 7

4.3. The Integral and Comparison Tests

4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece