please dont just give letter answer, any little explaination would be very helpf
ID: 2841749 • Letter: P
Question
please dont just give letter answer, any little explaination would be very helpful!
Suppose an = ln(n3)/n5/4 and bn = 1/n5/4 and you apply the direct comparison test to the series with prototype series . Which of the following statements regarding the series is true? The series diverges by the direct comparison test with the given prototype series. The series converges by the nth-term test. The series converges by the direct comparison test with the given prototype series. The series diverges by the nth-term test. Both the nth-term test and the direct comparison test with the given prototype series are inconclusive.Explanation / Answer
b(n) = 1/n^(5/4) is a convergent p-series
If you do a direct comparison test with a convergent comparison series, you need a(n) <= b(n) to be true.
ln(n^3)/n^(5/4) <= 1/n^(5/4)
The bottoms are the same, so really you are concerned with ln(n^3) <= 1. This isn't always true since range of ln(x) is (-inf, inf). So direct comparison test is inconclusive if we compare with 1/n^(5/4)
In nth term test we take this limit:
lim n--> inf a(n)
If that limit is not 0, then we can say the series diverges. If it is 0, that is inconclusive.
lim n--> inf ln(n^3) / n^(5/4) gives inf/inf by direct sub so you can use l'hospital's rule
lim n--> inf (3n^2/n^3) / ((5/4)n^(1/4))
lim n--> inf (3/n) / [(5/4)n^(1/4)]
lim n--> inf (12/5) / n^(5/4) = 0
Since lim n--> inf a(n) = 0, the nth term test is also inconclusive.
The answer is E: Both direct comparison and nth term test are inconclusive.
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