Question
Asked Dec 4, 2019
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IV. 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
+
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IV. 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 +

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Expert Answer

Step 1

 (Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question specifying the one you’d like answered)

Root test is the test for the positive term series. It states that,

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Leta, 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.

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Step 2

Ratio test for positive term series:

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Leta, 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

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Step 3

Integral te...

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Leta 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.

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