Video: Applying the 𝑛th Term Divergence Test with a Trigonometric Function

What can we conclude by applying the 𝑛th term divergence test in the series βˆ‘ _(𝑛 = 1) ^(∞) 3 cos 𝑛?

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Video Transcript

What can we conclude by applying the 𝑛th term divergence test in the series the sum from 𝑛 equals one to ∞ of three cos 𝑛?

Let’s begin by recalling what the 𝑛th term divergence test tells us. It tells us that if the limit as 𝑛 approaches ∞ of π‘Ž sub 𝑛 does not exist or if it’s not equal to zero, then the series the sum from 𝑛 equals one to ∞ of π‘Ž sub 𝑛 is divergent. We also recall that if the limit is equal to zero, we can’t tell whether the series converges or diverges. And so we say that the test fails or it’s inconclusive.

In this question, we can define π‘Ž sub 𝑛 to be equal to three cos 𝑛. And we’re going to need to evaluate the limit as 𝑛 approaches ∞ of three cos 𝑛. Now actually, there’s not a lot we need to do to evaluate this limit. Let’s sketch the graph of three cos 𝑛 and see if we can recall what it looks like.

The graph of 𝑦 equals three cos π‘₯ is shown. We can see that it’s an oscillating function. It has peaks and troughs at 𝑦-values of three and negative three. This continues forever. And so we say that the limit as 𝑛 approaches ∞ of three cos 𝑛 does not exist. We cannot pinpoint it as one particular number or even ∞.

And by the 𝑛th term divergence test, that tells us that the series the sum from 𝑛 equals one to ∞ of three cos 𝑛 is divergent. It diverges.

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