Lecture 26 : Comparison Test

1 Comparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleComparison TestIn this section, as we did with improper integrals, we see how to compare aseries (with Positive terms) to a well known series to determine if it convergesor will of course make use of our knowledge ofp-series and geometricseries. Xn=11npconverges forp>1,diverges forp 1. Xn=1arn 1converges if|r|<1,diverges if|r| TestSuppose thatPanandPbnare serieswith positiveterms.(i) IfPbnis convergent andan bnfor alln, thanPanis alsoconvergent.(ii)IfPbnis divergent andan bnfor all n, thenPanis PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleComparison TestIn this section, as we did with improper integrals, we see how to compare aseries (with Positive terms) to a well known series to determine if it convergesor will of course make use of our knowledge ofp-series and geometricseries.

2 Xn=11npconverges forp>1,diverges forp 1. Xn=1arn 1converges if|r|<1,diverges if|r| TestSuppose thatPanandPbnare serieswith positiveterms.(i) IfPbnis convergent andan bnfor alln, thanPanis alsoconvergent.(ii)IfPbnis divergent andan bnfor all n, thenPanis PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleComparison TestIn this section, as we did with improper integrals, we see how to compare aseries (with Positive terms) to a well known series to determine if it convergesor will of course make use of our knowledge ofp-series and geometricseries. Xn=11npconverges forp>1,diverges forp 1. Xn=1arn 1converges if|r|<1,diverges if|r| TestSuppose thatPanandPbnare serieswith positiveterms.

3 (i) IfPbnis convergent andan bnfor alln, thanPanis alsoconvergent.(ii)IfPbnis divergent andan bnfor all n, thenPanis PilkingtonLecture 26 : Comparison TestComparison TestExample 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 1 Example 1 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=12 1/nn3 IFirst we check thatan>0 >true since2 1/nn3>0 forn have 21/n=n 2>1 forn 1. Therefore 2 1/n=1n 2<1 forn 1/nn3<1n3forn> n=11n3is a p-series withp>1, it the above series withP n=11n3, we can conclude thatP n=12 1/nn3also converges andP n=12 1/nn3 P n=11n3 Annette PilkingtonLecture 26 : Comparison TestComparison TestExample 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 1 Example 1 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=12 1/nn3 IFirst we check thatan>0 >true since2 1/nn3>0 forn have 21/n=n 2>1 forn 1.

4 Therefore 2 1/n=1n 2<1 forn 1/nn3<1n3forn> n=11n3is a p-series withp>1, it the above series withP n=11n3, we can conclude thatP n=12 1/nn3also converges andP n=12 1/nn3 P n=11n3 Annette PilkingtonLecture 26 : Comparison TestComparison TestExample 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 1 Example 1 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=12 1/nn3 IFirst we check thatan>0 >true since2 1/nn3>0 forn have 21/n=n 2>1 forn 1. Therefore 2 1/n=1n 2<1 forn 1/nn3<1n3forn> n=11n3is a p-series withp>1, it the above series withP n=11n3, we can conclude thatP n=12 1/nn3also converges andP n=12 1/nn3 P n=11n3 Annette PilkingtonLecture 26 : Comparison TestComparison TestExample 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 1 Example 1 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=12 1/nn3 IFirst we check thatan>0 >true since2 1/nn3>0 forn have 21/n=n 2>1 forn 1.

5 Therefore 2 1/n=1n 2<1 forn 1/nn3<1n3forn> n=11n3is a p-series withp>1, it the above series withP n=11n3, we can conclude thatP n=12 1/nn3also converges andP n=12 1/nn3 P n=11n3 Annette PilkingtonLecture 26 : Comparison TestComparison TestExample 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 1 Example 1 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=12 1/nn3 IFirst we check thatan>0 >true since2 1/nn3>0 forn have 21/n=n 2>1 forn 1. Therefore 2 1/n=1n 2<1 forn 1/nn3<1n3forn> n=11n3is a p-series withp>1, it the above series withP n=11n3, we can conclude thatP n=12 1/nn3also converges andP n=12 1/nn3 P n=11n3 Annette PilkingtonLecture 26 : Comparison TestComparison TestExample 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 1 Example 1 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=12 1/nn3 IFirst we check thatan>0 >true since2 1/nn3>0 forn have 21/n=n 2>1 forn 1.

6 Therefore 2 1/n=1n 2<1 forn 1/nn3<1n3forn> n=11n3is a p-series withp>1, it the above series withP n=11n3, we can conclude thatP n=12 1/nn3also converges andP n=12 1/nn3 P n=11n3 Annette PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 2 Example 2 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=121/nnIFirst we check thatan>0 >true since21/nn>0 forn have 21/n=n 2>1 forn >1nforn> n=11nis a p-series withp= 1 ( the harmonic series), , by Comparison , we can conclude thatP n=121/nnalso PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 2 Example 2 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=121/nnIFirst we check thatan>0 >true since21/nn>0 forn have 21/n=n 2>1 forn >1nforn> n=11nis a p-series withp= 1 ( the harmonic series), , by Comparison , we can conclude thatP n=121/nnalso PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 2 Example 2 Use the Comparison test to determine if the following seriesconverges or diverges.

7 Xn=121/nnIFirst we check thatan>0 >true since21/nn>0 forn have 21/n=n 2>1 forn >1nforn> n=11nis a p-series withp= 1 ( the harmonic series), , by Comparison , we can conclude thatP n=121/nnalso PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 2 Example 2 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=121/nnIFirst we check thatan>0 >true since21/nn>0 forn have 21/n=n 2>1 forn >1nforn> n=11nis a p-series withp= 1 ( the harmonic series), , by Comparison , we can conclude thatP n=121/nnalso PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 2 Example 2 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=121/nnIFirst we check thatan>0 >true since21/nn>0 forn have 21/n=n 2>1 forn >1nforn> n=11nis a p-series withp= 1 ( the harmonic series), , by Comparison , we can conclude thatP n=121/nnalso PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 2 Example 2 Use the Comparison test to determine if the following seriesconverges or diverges.

8 Xn=121/nnIFirst we check thatan>0 >true since21/nn>0 forn have 21/n=n 2>1 forn >1nforn> n=11nis a p-series withp= 1 ( the harmonic series), , by Comparison , we can conclude thatP n=121/nnalso PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 3 Example 3 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=11n2+ 1 IFirst we check thatan>0 >true since1n2+1>0 forn haven2+ 1>n2forn +1<1n2forn> n=11n2is a p-series withp= 2, it , by Comparison , we can conclude thatP n=11n2+1also convergesandP n=11n2+1 P n= PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 3 Example 3 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=11n2+ 1 IFirst we check thatan>0 >true since1n2+1>0 forn haven2+ 1>n2forn +1<1n2forn> n=11n2is a p-series withp= 2, it , by Comparison , we can conclude thatP n=11n2+1also convergesandP n=11n2+1 P n= PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 3 Example 3 Use the Comparison test to determine if the following seriesconverges or diverges.

9 Xn=11n2+ 1 IFirst we check thatan>0 >true since1n2+1>0 forn haven2+ 1>n2forn +1<1n2forn> n=11n2is a p-series withp= 2, it , by Comparison , we can conclude thatP n=11n2+1also convergesandP n=11n2+1 P n= PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 3 Example 3 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=11n2+ 1 IFirst we check thatan>0 >true since1n2+1>0 forn haven2+ 1>n2forn +1<1n2forn> n=11n2is a p-series withp= 2, it , by Comparison , we can conclude thatP n=11n2+1also convergesandP n=11n2+1 P n= PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 3 Example 3 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=11n2+ 1 IFirst we check thatan>0 >true since1n2+1>0 forn haven2+ 1>n2forn +1<1n2forn> n=11n2is a p-series withp= 2, it , by Comparison , we can conclude thatP n=11n2+1also convergesandP n=11n2+1 P n= PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 3 Example 3 Use the Comparison test to determine if the following seriesconverges or diverges.

10 Xn=11n2+ 1 IFirst we check thatan>0 >true since1n2+1>0 forn haven2+ 1>n2forn +1<1n2forn> n=11n2is a p-series withp= 2, it , by Comparison , we can conclude thatP n=11n2+1also convergesandP n=11n2+1 P n= PilkingtonLecture 26 : Comparison TestComparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example ExampleExample 4 Example 4 Use the Comparison test to determine if the following seriesconverges or diverges: Xn=1n 22nIFirst we check thatan>0 >true sincen 22n=1n22n>0 forn have1n22n<1n2forn n=11n2i