Video Transcript
Use the 𝑛th term divergence test
to determine whether the series the sum from 𝑛 equals one to infinity of three 𝑛
over five 𝑛 is divergent or whether the test is inconclusive.
Let’s begin by recalling what we
actually know about the 𝑛th term divergence test. This tells us that if the limit as
𝑛 approaches infinity 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 infinity of 𝑎 sub 𝑛 is
divergent. And now whilst at series that
converges we’ll have a limit that is equal to zero. If we’re just working with the
limit, and that’s equal to zero, we can’t actually tell whether the series converges
or diverges. And we say that the test fails or
it’s inconclusive.
And so we let 𝑎 sub 𝑛 be equal to
three to the 𝑛th power over five to the 𝑛th power. And so we want to find the limit as
𝑛 approaches infinity of three to the 𝑛th power over five to the 𝑛th power. Now if we were to substitute in 𝑛
equals infinity, we just end up with infinity over infinity, which we know to be
undefined. And so we’re going to manipulate
our expression somewhat. We’re going to write it as the
limit as 𝑛 approaches infinity of three-fifths to the 𝑛th power.
Now this is really useful because
three-fifths is a fraction which is less than one. And so if we have 𝑛 is equal to
two, we get three-fifths all squared, which is nine over 25. That’s less than three-fifths. If 𝑛 is equal to three, we have
three-fifths cubed, which is equal to 27 over 125, which is less than nine over
25.
Each value of 𝑛 makes the fraction
three-fifths to the 𝑛th power ever smaller. And that actually means that the
limit as 𝑛 approaches infinity of three-fifths to the 𝑛th power must be zero. We recall that this doesn’t
actually tell us anything about whether the series converges or diverges. It fails or it’s inconclusive.
