13–52. Limits of sequences
Find the limit of the followi...

Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{2ⁿ⁺¹3⁻ⁿ}

Accepted Answer

Welcome back, everyone. Evaluate the limit of the sequence B N equals 5 of N multiplied by 4 to the power of negativeN + 2 as N approaches infinity or determine if it diverges. A 0 B 5 divided by 4, C 20, and D diverges. So for this problem, we want to begin by stating the limit as an approaches infinity of B subscript N. Which is 5, raised the power of N multiplied by 4, raises the power of negative N + 2. What we can do is simplify this expression by applying the properties of exponents. We're going to get limit as an approaches infinity. Of 5 raises the power of N multiplied by 4 raises the power of negative m. Which is multiplied by 4 squared, right? Basically, we're applying the properties of exponents and splitting our exponent negative N + 2 into a product with two separate exponents. Now we can continue and we can show that this is limits and approaches infinity of. 5 to the power of n multiplied by 14th raises the power of N. Because we have a negative and, we can essentially use the reciprocal rule for fractions and show that 4 raises the power of -1, right? Because this is 4 raises the power of -1 or raise the power of N, we can basically convert it into 14th raise the power of N. And then we're multiplying by 4 squad, which is 16. Let's go ahead and factor out our constant, which is 16. Multiply it by the resultant limit as an approaches infinity of. 5 to the power of n multiplied by 1/4 to the power of n can be written as 5/4 raised to the power of n. And now we can analyze the behavior as N approaches infinity. Notice that 5 divided by 4 is greater than 1, and if we are raising a number that is greater than 1 to an infinitely large number, that number is going to approach infinity, right? So we're going to get 16 multiplied by infinity, which is just infinity. And now because our limit is infinite, we can conclude that this sequence diverges, right? Well then we have our final answer and thank you for watching.

13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{2ⁿ⁺¹3⁻ⁿ}

Key Concepts

Limits of Sequences

Exponential Functions and Their Behavior

Simplifying and Manipulating Sequences

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