Using the Root Test
In Exercises 9–16, use the Root Test...

Question

Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²]

(Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)

Accepted Answer

Welcome back everyone. Apply the root test to determine whether the series sigma from n equals 3 up to infinity of 1 minus 3 divided by n to the power of n 2 converges absolutely or diverges. Hint limit as n approaches infinity of 1 plus x divided by n to the power of n is equal to e to the power of x. A converges absolutely. And b diverges. So for this problem we want to identify the limit l of the nth term. Specifically, we want to identify the limit as n approaches infinity of our nth term is 1 minus 3 divided by n. To the power of n 2, and since we're applying the root test, we also want to take the nth root of the nth term, right? So let's go ahead and add the root. And now let's go ahead and simplify. So this is going to be the limit. As n approaches infinity of 1 minus 3 divided by n to the power of n 2 raised to the power of 1 divided by n, this is how we can change the radical into an exponent. And now let's use the properties of exponents. We get limit as n approaches infinity of 1 minus 3 divided by n to the power of n 2 divided by n, which is simply n. And now we're going to use the hand. So what we can do is simply rewrite it as a limit as an approaches infinity of 1 plus. 3 divided by N. Erase the power of n. Notice that x is equal to -3 in the given identity, so the limit is equal to e to the power of -3, which is 1 divided by e cubed. According to the root test, if the limit is equal, if the limit is less than 1, the series converges absolutely, and this is the case here. L is equal to 1 divided by e cubed, which is less than 1. So the series converges absolutely, and the correct answer is option A. Thank you for watching.

Using the Root TestIn Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.∑(from n=1 to ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²](Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)

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Using the Root Test
In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.
∑(from n=1 to ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²]
(Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)