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Answer :
Final answer:
Based on the inconclusive results from both the Ratio Test and the Root Test, we cannot determine whether the series is convergent or divergent.
Explanation:
To determine whether the series is convergent or divergent, we will use the Ratio Test and the Root Test.
Ratio Test:
The Ratio Test states that if the limit of the absolute value of the ratio of consecutive terms in a series is less than 1, then the series converges. If the limit is greater than 1 or does not exist, the series diverges.
Let's apply the Ratio Test to the given series:
149 n1 (n1)820 + 1
First, let's find the ratio of consecutive terms:
r = (n+1)^(820+1) / n^(820)
Next, let's take the limit as n approaches infinity:
lim (n->∞) |r| = lim (n->∞) |(n+1)^(820+1) / n^(820)|
Since the limit is not explicitly given in the question, we cannot determine whether it is less than 1 or greater than 1. Therefore, the Ratio Test is inconclusive for this series.
Root Test:
The Root Test states that if the limit of the nth root of the absolute value of the terms in a series is less than 1, then the series converges. If the limit is greater than 1 or does not exist, the series diverges.
Let's apply the Root Test to the given series:
Σε(1 + 1)" 3
First, let's find the nth root of the absolute value of the terms:
r = (1 + 1)^(1/3)
Next, let's take the limit as n approaches infinity:
lim (n->∞) |r| = lim (n->∞) |(1 + 1)^(1/3)|
Since the limit is not explicitly given in the question, we cannot determine whether it is less than 1 or greater than 1. Therefore, the Root Test is inconclusive for this series as well.
Based on the inconclusive results from both the Ratio Test and the Root Test, we cannot determine whether the series is convergent or divergent.
Learn more about convergence and divergence of series here:
https://brainly.com/question/33953891
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