12–24. Limits of sequences Evaluate the limit of the sequence ...

Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (–1)ⁿ / 0.9ⁿ

Accepted Answer

Welcome back, everyone. Find the limit of the sequence given by CN equals -3 raised to the power of N divided by 0.7 raises to the power of N. SN approaches infinity or determine that the sequence diverges. A 0 B 1 C 30 divided by 7, and D diverges. For this problem, let's go ahead and write our limit. Limit as an approaches infinity. Of Cn, which is a limit as n approaches infinity of -3 raises the power of n divided by 0.7 raises the power of n. We can identify this limit by applying the properties of exponents. Let's go ahead and write limit as an approaches infinity. Starting with -3, raise the power of N, we can write it as -1 multiplied by 3, raise the power of N. And we're dividing by 0.7 rates the power of N. Now, considering our numerator, we can rewrite it as -1, raise the power of N, multiplied by 3, raise the power of N. Divided by 0.7 raises the power of n. And now let's combine our terms, 3 to the power of N and 0.7 to the power of N. Because we have different bases, but the same exponent, we can write that expression as -1 raise the power of N first, and then 3 divided by 0.7 or raise the power of N. What we can do is simply rewrite it as a fraction with whole numbers. We're going to multiply both sides by 10, and we get limit as N approaches infinity of -1, raises the power of N multiplied by 30 divided by 7, raise the power of n. Now notice that as N approaches infinity, we are raising 30 divided by 7 to the power of N, right? because our number is greater than 1 and we're raising it to an infinitely large exponent, it is going to approach infinity. Considering -1 raises the power of N, it oscillates between -1 and 1. So our limit oscillates between positive infinity and negative infinity, right? It grows without bounds in both directions, which means that the limit does not exist. If the limit, the limit of CM does not exist, it essentially means that our sequence diverges and the correct answer for this problem corresponds to the answer choices D. Thank you for watching.

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ / 0.9ⁿ

Key Concepts

Limit of a Sequence

Behavior of Exponential Functions

Alternating Sequences

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