Question Video: Deciding If an Alternating Harmonic Series Is Absolutely Convergent, Conditionally Convergent, or Divergent Mathematics • Higher Education

Video Transcript

Is the alternating harmonic series the sum from 𝑛 equals one to ∞ of negative one to the power of 𝑛 add one multiplied by one over 𝑛 absolutely convergent, conditionally convergent, or divergent?

Let’s firstly remember that a series is absolutely convergent if the series of absolute values is convergent. And a series is conditionally convergent if the series of absolute values diverges but the series converges. And otherwise, the series is divergent. So let’s start by testing for absolute convergence. We can see that negative one raised to the power of 𝑛 add one is always going to give us one when we take the absolute value. So this is in fact the same as the sum from 𝑛 equals one to ∞ of one over 𝑛. But this is actually a series that we’re familiar with. It’s the harmonic series. And we know that the harmonic series diverges. So the alternating harmonic series is not absolutely convergent. But is it conditionally convergent or divergent?

So our next step is to test the alternating harmonic series for convergence. Because that’s an alternating series, we can do this with the alternating series test. Recall this says that for an alternating series, the sum of negative one to the power of 𝑛 add one multiplied by 𝑏 𝑛 if 𝑏 𝑛 is decreasing and the limit as 𝑛 approaches ∞ of 𝑏 𝑛 is equal to zero, then 𝑎 𝑛 is convergent. So for the alternating harmonic series, we can say that 𝑏 𝑛 equals one over 𝑛. So is 𝑏 𝑛 decreasing? Well, as 𝑛 increases, one over 𝑛 does decrease. So that condition is satisfied. But does the limit as 𝑛 approaches ∞ of 𝑏 𝑛 equal zero? Well, the limit as 𝑛 approaches ∞ of one over 𝑛 is going to be one over ∞, which we know is zero. So that condition is satisfied. So because we found that the alternating harmonic series is not absolutely convergent, but it is convergent, we can conclude that the alternating harmonic series is conditionally convergent.

We can summarize the check for absolute convergence, conditional convergence, and divergence in a helpful diagram. Let’s say we want to find out whether the series 𝑎 𝑛 is absolutely convergent, conditionally convergent, or divergent. We begin by testing whether the series of absolute values is convergent or divergent. Let’s say that we find that the series of absolute values is convergent. Then, the series 𝑎 𝑛 is absolutely convergent. But if we find that the series of absolute values is divergent, then the series 𝑎 𝑛 is not absolutely convergent. But it may still be conditionally convergent. So we try a different test on the series 𝑎 𝑛 to check for convergence, for example, the alternating series test.

And if we find that the series 𝑎 𝑛 converges, then we say that the series 𝑎 𝑛 is conditionally convergent. But if we find that the series 𝑎 𝑛 diverges, then we conclude that the series 𝑎 𝑛 is divergent. So these are the three possible conclusions that we can draw.