What can we conclude by applying
the 𝑛th term divergence test in the series the sum from 𝑛 equals one to ∞ of three
Let’s begin by recalling what the
𝑛th term divergence test tells us. It tells us that if the limit as 𝑛
approaches ∞ of 𝑎 sub 𝑛 does not exist or if it’s not equal to zero, then the
series the sum from 𝑛 equals one to ∞ of 𝑎 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 𝑎
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 𝑥
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.