# IV. Evaluate using root test, ratio test and integral test (allthree). State why a series is qualified or not for one of thetestsVn-1п8 п-0In(n9. п-0210.Tn5n6+

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1 views help_outlineImage TranscriptioncloseIV. Evaluate using root test, ratio test and integral test (all three). State why a series is qualified or not for one of the tests Vn-1 п 8 п-0 In(n 9. п-0 2 10. Tn5n6 + fullscreen
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Step 1

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Root test is the test for the positive term series. It states that, help_outlineImage TranscriptioncloseLeta, be a positive term series. Then the series (a) Converges if lim(a )» <1. (b) Diverges if lim(a,) > n 1. (c) Fails if lim(a)n = 1 Given Vn- Ση. п n-1 л -1 Let a п n-1 Vn-1 lim 2 lim (a п п-1 lim "(/n+1) n 1 lim 1 1 п = 1 Hence, root test fails here. fullscreen
Step 2

Ratio test for positive term series: help_outlineImage TranscriptioncloseLeta, be a positive term series. Then the series аp1 <1. (a) Converges if lim а, (b) Diverges if lim = 1 (c) Fails if lim noa Vn1 X аp1 lim п = lim n-o an n+1 п -1 Vn1- X п lim n+1 1 1+ 1 1 = lim X 1 1 1 1+ no n п = 1 Hence, ratio test also fails here fullscreen
Step 3

Integral te... help_outlineImage TranscriptioncloseLeta bea positive, monotonically decreasing series. Then the series converges or diverges iff the integral f(x)dx converges or diverges. Here, the given series is monotonically increasing. Hence, integral test fails here. fullscreen

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