would love the understand thought process! Suppose an = 7n + 8n/5n + 6n and bn =
ID: 2841718 • Letter: W
Question
would love the understand thought process!
Suppose an = 7n + 8n/5n + 6n and bn = (8/5)n and you apply the limit comparison test to the series with prototype series . Which of the following statements regarding the series is true? Both the nth-term test and the limit comparison test with the given prototype series are inconclusive. The series converges by the limit comparison test with the given prototype series. The series diverges by the limit comparison test with the given prototype series. The series converges by the nth-term test. The series diverges by the nth-term test.Explanation / Answer
b(n) = (8/5)^n is a divergent geometric series. (Note this can also be written as 8^n / 5^n.)
If you want to apply limit comparison test, you want to find:
lim n--> inf a(n) / b(n)
lim n--> inf [(7^n + 8^n) / (5^n + 6^n)] / [8^n / 5^n]
lim n--> inf [(7^n + 8^n) / (5^n + 6^n] * [5^n / 8^n]
Since lim n--> inf, only the largest exponential matter in a(n):
lim n--> inf [8^n / 6^n] * [5^n / 8^n]
lim n--> inf [5^n / 6^n] = 0 since the largest exponential is below.
Since the limit exists, this means the given series a(n) can behave like the known series b(n) and diverges.
nth term test refers to this test:
lim n--> inf [a(n)]
If this limit does not go to 0, then the series diverges.
lim n--> inf (7^n + 8^n) / (5^n + 6^n)
lim n--> inf 8^n / 6^n = infinity, since the highest exponential is on top
Since the limit does not go to 0, the series diverges by nth term test as well.
In summary, a(n) diverges by both the limit comparison test and the nth term test.
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