True or False: If 𝑎 𝑛 approaches
zero as 𝑛 approaches ∞, then the 𝑛th term divergence test fails for the series the
sum from 𝑛 equals zero to ∞ of 𝑎 𝑛.
We’re thinking about a series in
which 𝑎 𝑛 approaches zero as 𝑛 approaches ∞. An example of such a series would
be the harmonic series, the sum from 𝑛 equals one to ∞ of one over 𝑛. Here, 𝑎 𝑛 is equal to one over
𝑛. And as 𝑛 gets bigger and
approaches ∞, one over 𝑛 will get smaller and approach zero.
So that’s an example of what this
series could be. But let’s start by reminding
ourselves of what the 𝑛th term divergence test says.
The 𝑛th term divergence test says
that if the limit as 𝑛 approaches ∞ of 𝑎 𝑛 does not exist or is not equal to
zero, then the series the sum from 𝑛 equals zero to ∞ of 𝑎 𝑛 is divergent. The really important thing to take
away from this definition though is that if the limit is equal to zero, we don’t
know anything about the convergence or divergence of the series. Just because the series doesn’t fit
this criteria doesn’t mean that the series is convergent. It could still be divergent. And actually, this is the case of
the harmonic series. We found the limit to be zero. But by another test, we can prove
that in fact the harmonic series is divergent.
So when the limit is zero, we can’t
draw any conclusions about the convergence or divergence of the series. Instead, we just say that the test
fails, and we would have to choose a different test. So this statement is true.