COMPARISON TESTS. Determine whether the following series converges or diverges n : n=1 : 4. 4 n = n < 1

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Russell Webster
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1 MTH 6 COMPARISON TESTS Compariso Test for Covergece Example. Solutio. For every, Sice P = X = + + < = is a coverget geometric series (for r = < ), the give series coverges by the Compariso Test for Covergece. Example. Solutio. For every, Sice P = = for Covergece. Example. X = is a coverget p-series (for p = p + p + < p = = > ), the give series coverges by the Compariso Test X = + ( + ) Solutio. For every, + ( + ) < + ( = 8 ) = 8 ; because we icrease the umerator ad decrease the deomiator. Sice P series (for r = < ), the give series coverges by the Compariso Test for Covergece. Example. Solutio. For every, = X = + p 7= p + < = 7 7= = = because we icrease the umerator ad decrease the deomiator. Sice P = is a coverget geometric = (for p = > ), the give series coverges by the Compariso Test for Covergece. is a coverget p-series

2 Example. X = l () = Solutio. It should be oted that l (x) < x for x > 0 (see the gure below). For every, Sice P = = for Covergece. is a coverget p-series (for p = Compariso Test for Divergece l () = < = = = > ), the give series coverges by the Compariso Test Example 6. Solutio. For every, Sice P = Divergece. = X = p p > p = = is a diverget p-series (for p = < ), the give series diverges by the Compariso Test for Example 7. X = p + Solutio. For every, p p p + > + = = =

3 Sice P = Divergece. = is a diverget p-series (for p = < ), the give series diverges by the Compariso Test for Example 8. Solutio. For every, Sice P = Divergece. = X = p + p + > p + = p = 8 = is a diverget p-series (for p = < ), the give series diverges by the Compariso Test for Example 9. X = + Solutio. For every, + < + = so that + > Sice P = + > = > ad thus is the diverget harmoic series, the give series diverges by the Compariso Test for Divergece. Example 0. X = + si () p + Solutio. Note that si () 0 for all. For every, + si () p > p p = + + = because we decrease the umerator ad icrease the deomiator. Sice P p = < ), the give series diverges by the Compariso Test for Divergece. Limit Compariso Test Example. Solutio. = = X = e ( + ) We shall apply the Limit Compariso Test with is a diverget p-series (for Sice a! b! a = e ( + ) ad b = e = e (+) e = lim! ( + ) = lim! e + = 0 ( + 0) = > 0

4 ad P = e is a coverget geometric series (for r = e < ), the give series also coverges Example. the Solutio. Sice P Test. = If we let a! b! X = a = p p = lim p! p ad b = ; = lim! q = p 0 = > 0 is a coverget p-series (for p = > ), the give series also coverges by the Limit Compariso Example. X si = Solutio. If we let a = si the it follows from the l Hôpital s Rule that a si = lim! b!! Sice P Test. = cos ad b = ; = lim cos! = cos (0) = > 0 is the diverget harmoic series, the give series also diverges i view of the Limit Compariso Example. X = l + Solutio. Cosider By virtue of the l Hôpital s Rule, Sice P = a! b! l + a = l + = lim! ad b = [ l ()] + = lim l ()! + = + 0 = > 0 is a coverget geometric series (for r = < ), the give series also coverges i view of the Limit Compariso Test. Example. X = +

5 Solutio. Cosider By virtue of the l Hôpital s Rule, a = + ad b = a! b! + = lim! + = lim! + = + 0 = > 0 P Sice is diverget geometric series (for r = > ), the give series also diverges i view of the Limit = Compariso Test.

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