9–30. The Ratio and Root Tests Use the Ratio Test or the Root ...

Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

1 + (1 / 2)² + (1 / 3)³ + (1 / 4)⁴ + ⋯

Accepted Answer

Welcome back, everyone. Determine whether the series sigma from N equals 1 up to infinity of 1 divided by 2 ends the power of N converges absolutely or diverges using the root test. Let's recall for this problem that whenever we apply the root test, we want to identify L where L is equal to the limits and approaches infinity of the n truth of the absolute value of a subscript n. So let's go ahead and evaluate the limit as an approach infinity of the untruth of. The absolute value of 1 divided by 2 and erase the power of n. Now for simplicity, we can rewrite our term A subscript M. As one. Divided by 2 M All erased to the power of n. We can do that because one raises the power of n is always one regardless of the value of n, right? And we can also drop the absolute value because 2N is always positive in this case. We begin with N equals 1. So the minimum value is going to be 2. And all of the remaining values, starting with N of 23, and so on, are going to be positive. So we can also remove the absolute value and now notice that. We can cancel out our exponent because we're taking and truth of the nth exponent. And this gives us a limit. As M approaches infinity. Of 1 divided by 2 and. Notice that our numerator has a constant and our denominator grows infinitely, right? As N approaches infinity, our denominator is going to approach infinity. So whenever we divide a constant by an infinitely large number, the limit is going to be 0. So we get a limit of 0. Meaning L is 0, and according to the root test, we have to compare our limit with 1. If our limit is less than 1, which is the case here, then the series converges absolutely. So the answer to this problem. Would it be convergence absolutely. That's our final answer for this problem, and thank you for watching.

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
1 + (1 / 2)² + (1 / 3)³ + (1 / 4)⁴ + ⋯

Key Concepts

Ratio Test

Root Test

Absolute Convergence

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