Question Video: Applying the Nth Term Divergence Test to a Series Mathematics

Video Transcript

What can we conclude by applying the 𝑛th of term divergence test in the series the sum from 𝑛 equals one to ∞ of two 𝑛 over six 𝑛 squared add four.

Let’s take a look at the 𝑛th term divergence test. This says that if the limit as 𝑘 approaches ∞ of 𝑎𝑘 is not equal to zero, then the sum of 𝑎𝑘 diverges. If the limit as 𝑘 approaches ∞ of a𝑘 equals zero, then the sum of 𝑎𝑘 may or may not converge. So, what this is telling us is that we need to find the limit as 𝑛 approaches ∞ of two 𝑛 over six 𝑛 squared add four.

To do this, we remember that when we take a limit at ∞ for a polynomial, all we need to do is look at the term with the largest power. The degree of the numerator is one. And the degree of the denominator is two. And so, because the numerator has a smaller degree than the denominator, this limit is equal to zero. So, we’re in this scenario, since we found the limit to be zero. So, this may or may not converge. Therefore, the divergence test is inconclusive.