Video Transcript
The alternating series test does
not apply to the series the sum of 𝑛 equals one to ∞ of negative one to the 𝑛th
power multiplied by 𝑛 over 𝑛 add one. What is the reason?
Let’s begin by reviewing the
alternating series test and the conditions required in order to use it. The alternating series test says
that for an alternating series, the sum from 𝑛 equals one to ∞ of negative one to
the 𝑛th power multiplied by 𝑎 𝑛, where 𝑎 𝑛 is greater than zero for all 𝑛,
then if, one, the limit as 𝑛 approaches ∞ if 𝑎 𝑛 is equal to zero and, two, the
sequence 𝑎 𝑛 is a decreasing sequence, then the series the sum from 𝑛 equals one
to ∞ of negative one to the 𝑛th power multiplied by 𝑎 𝑛 is a convergent
series.
So, we can see here there’s a few
conditions we must satisfy before we can draw any conclusions from the alternating
series test. Firstly, and most importantly, the
series must be alternating. We must have that 𝑎 𝑛 is greater
than zero for all 𝑛, the limit as 𝑛 approaches ∞ if 𝑎 𝑛 is equal to zero, and
the sequence 𝑎 𝑛 is a decreasing sequence.
So, first of all, do we have an
alternating series? Well, the negative one raised to
the 𝑛th power creates this alternating effect. 𝑛 runs through positive
numbers. And we’re going to find that
negative one to the 𝑛th power alternates between one and negative one. We could also look at the first few
terms of this series. And we can see that these are
alternating between positive and negative terms.
We must also have that 𝑎 𝑛 is
greater than zero. 𝑎 𝑛 is the bit being multiplied
by the negative one to the 𝑛th power. So, for this series, 𝑎 𝑛 is 𝑛
over 𝑛 add one, and this needs to work for all 𝑛. In this case, 𝑛 only runs through
positive values. And we find that if we substitute
any positive value of 𝑛, 𝑛 over 𝑛 add one is always greater than zero, so that’s
okay.
Now, before we can say anything
from the alternating series test, we’ve got to check that the limit as 𝑛 approaches
∞ of 𝑎 𝑛 is equal to zero. This means checking whether the
limit as 𝑛 approaches ∞ of 𝑛 over 𝑛 add one is equal to zero. So, how are we going to do
this?
Let’s start by dividing through all
the terms by 𝑛, and we find that 𝑛 over 𝑛 just cancels to one. Now, as 𝑛 approaches ∞, we find
that one over 𝑛 approaches zero, in which case the whole limit approaches one over
one, which is in fact just one.
So, we found that the limit as 𝑛
approaches ∞ of 𝑎 𝑛 is not equal to zero. So, this is the reason why the
alternating series test does not apply to this series. It’s because the limit as 𝑛
approaches ∞ of 𝑛 over 𝑛 add one is not equal to zero.
