55–70. More sequences
Find the limit of the following se...

Question

55–70. More sequences

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

aₙ = e⁻ⁿcosn

Accepted Answer

Welcome back, everyone. Determine the limit of the sequence DN equals 5 of N divided by 7 power of N, multiplied by cosine of 2 N Sn approaches infinity. A 1 B 0 C1 and D5 divided by 7. For this problem, let's go ahead and write down the limits and approaches infinity of D N. Which is a limit as N approaches infinity of 5 of N divided by 7 of N multiplied by cosine of 2N. What we can do to identify this limit is simply apply the properties of exponents. Specifically, we're going to get limit as N approaches infinity of 5 divided by 7 to the power of N because we have different bases but the same exponent. Multiplied by a cosine of 2 M. And now let's analyze this behavior as an approaches infinity. Now, cosine of 2 and basically oscillates between. -1 and 1, right, because that's the range of cosine. So when N goes to infinity, cosine just keeps oscillating between these two values. However, the most important term is 5 divided by 7 raises the power of N. Since N approaches infinity and 5 divided by 7 is less than 1, and specifically we want to show that. 5 divided by 7. Contains the absolute value, this is very important, right? Because we can also have a number that is negative, but greater than -1. This means that 5 divided by 7 weights the power of NS and approaches infinity is going to approach 0. Because we essentially have a numerator that is lower than the denominator, right? And as we Raise such a number to an infinitely large number, we're going to get 0 as our limit. So now 0 multiplied by a value that oscillates between -1 and 1 is going to be 0. So we can conclude that the correct answer for this problem corresponds to the answer choice B. Thank you for watching.

55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.

aₙ = e⁻ⁿcosn

Key Concepts

Limits of Sequences

Exponential Decay

Oscillatory Behavior of Trigonometric Functions

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