Video: Using the 𝑛th Term Divergence Test

Use the 𝑛th term divergence test to determine whether the series βˆ‘_(𝑛 = 1) ^(∞) (3^(𝑛)/5^(𝑛)) is divergent or whether the test is inconclusive.

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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 π‘Ž 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 π‘Ž 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 π‘Ž 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.

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