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 ∞) [4ⁿ / (3n)ⁿ]

Accepted Answer

Welcome back everyone. Use the root test to determine whether the series sigma from n equals 1 up to infinity of 5 to the power of n divided by 4 n to the power of n converges absolutely or diverges. A converges absolutely and b diverges. For this problem, let's remember the root that we want to identify the limit l. Which is limit as n approaches infinity of the nth root. Of the absolute value of a n. So in this case, we are evaluating the limit as an approach is infinity. Of the root. Of the absolute value of 5 to the power of n. Divided by 4 m to the power of n. Now what we're going to do is understand that an approach is infinity, and we have a ratio of two exponential functions with positive leading coefficients, right? So, essentially. We are going to drop the absolute value, so this is what we can do, right? Because our AM is. Always greater than 0 for n greater than or equal to 1 since we start at n equals 1 and then we're going to perform algebraic simplifications so this is going to be a limit as n approaches infinity. What we can do is rewrite ANS 5 divided by. Or n to the power of n using the properties of exponents, and we can also express the radical as 1 divided by n in the exponent. So now we can multiply the exponents to get limit as n approaches infinity of 5 divided by 4 n raise to the power of n multiplied by 1 divided by n, which basically gives us an exponent of 1. And now the limit is 0 since we're dividing a constant by an infinitely large number, 5 divided by 4, which is multiplied by infinity, right? Since the limit is equal to 0 according to the root test, if our limit is less than 1, the series converges absolutely. So 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 ∞) [4ⁿ / (3n)ⁿ]

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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 ∞) [4ⁿ / (3n)ⁿ]