Question Video: Determine Whether a Series Is Absolutely Convergent- Conditionally Convergent or Divergent Mathematics

Video Transcript

Consider the series the sum from 𝑛 equals one to ∞ of negative one to the 𝑛 add two power over four 𝑛 minus one. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

Let’s begin with a reminder of what it means for a series to be absolutely convergent or conditionally convergent. A series the sum of 𝑎 𝑛 is absolutely convergent if the series the sum of the absolute value of 𝑎 𝑛 is convergent. And it’s conditionally convergent if the series of absolute values diverges but the series itself still converges.

So let’s start by checking whether the series is absolutely convergent or not. This means testing for the convergence of the series the sum from 𝑛 equals one to ∞ of the absolute value of negative one to the 𝑛 add two power over four 𝑛 minus one. One thing to notice is that negative one to the 𝑛 add two power will always give us either one or negative one, depending on the value of 𝑛.

So taking the absolute value of negative one to the 𝑛 add two power will always give us one. Also, for values of 𝑛 greater than or equal to one, four 𝑛 minus one will always be positive. So we can actually rewrite this series as the sum from 𝑛 equals one to ∞ of one over four 𝑛 minus one. So we need to test this for convergence. We can do this using the comparison test, which tells us that for two series 𝑎 𝑛 and 𝑏 𝑛, both positive sequences with 𝑎 𝑛 less than or equal to 𝑏 𝑛, then if the series 𝑏 𝑛 converges, the series 𝑎 𝑛 also converges. But if the series 𝑎 𝑛 diverges, then the series 𝑏 𝑛 also diverges.

We could compare our series with the series from 𝑛 equals one to ∞ of one over four 𝑛. We can see that this is less than or equal to the series the sum from 𝑛 equals one to ∞ of one over four 𝑛 minus one. This works for all positive 𝑛. If we find that the series the sum from 𝑛 equals one to ∞ of one over four 𝑛 diverges, then we can conclude that our series also diverges. And actually, we can be rewrite the series the sum from 𝑛 equals one to ∞ of one over four 𝑛 by using the constant multiplication rule. This is just the same as one over four multiplied by the series the sum from 𝑛 equals one to ∞ of one over 𝑛. And this is a series that we recognize. It’s just the harmonic series. And we know this to be divergent. Therefore, the series that we’re looking at is also divergent by the comparison test.

So because the series of absolute values is divergent, the original series in the question is not absolutely convergent. Even though this series is not absolutely convergent, it may still be conditionally convergent. So that’s what we’ll check for next. So I’ll just clear some space to do that. To check for conditional convergence, we check whether the original series is convergent or not. One thing to notice about this series is that the negative one to the 𝑛 add two power creates an alternating effect between positive and negative values. So we can use the alternating series test.

The alternating series tests says that for a series the sum of 𝑎 𝑛, where 𝑎 𝑛 equals negative one to the 𝑛 add one power multiplied by 𝑏 𝑛, where 𝑏 𝑛 is greater than or equal to zero for all 𝑛, if the limit as 𝑛 approaches ∞ of 𝑏 𝑛 equals zero and the sequence 𝑏 𝑛 is a decreasing sequence, then the series 𝑎 𝑛 is convergent.

For this series, it’s the negative one to the 𝑛 add two power that creates the alternating effect. So our 𝑏 𝑛 is equal to one over four 𝑛 minus one. We need to check that as 𝑛 approaches ∞, this is going to give us zero and that this is a decreasing sequence. Well, as 𝑛 approaches ∞, four 𝑛 minus one will approach ∞. So the limit as 𝑛 approaches ∞ of one over four 𝑛 minus one is just zero.

We can also see that by looking at the first few terms of this sequence, this is a decreasing sequence. So we can conclude that the series the sum from 𝑛 equals one to ∞ of negative one to the 𝑛 add two power over four 𝑛 minus one is convergent. So we found the series to be convergent but not absolutely convergent. So the series is conditionally convergent.