Video: Using the Ratio Test

True or false: The series βˆ‘_(𝑛 = 1)^(∞) (1/(𝑛² + 1)) is convergent by the ratio test.

02:22

Video Transcript

True or false: The series, which is the sum from 𝑛 equals one to ∞ of one over 𝑛 squared plus one, is convergent by the ratio test.

We’ll need to apply the ratio test to this series. The ratio test tells us that for the series, which is the sum from 𝑛 equals one to ∞ of π‘Ž 𝑛, where 𝐿 is equal to the limit as 𝑛 tends to ∞ of the absolute value of π‘Ž 𝑛 plus one over π‘Ž 𝑛. Then, firstly, if 𝐿 is less than one, then the series converges absolutely. Secondly, if 𝐿 is greater than one, then the series diverges. And thirdly, if 𝐿 is equal to one, then the test is inconclusive.

From the series given in the question, we have that π‘Ž 𝑛 is equal to one over 𝑛 squared plus one. Therefore, π‘Ž 𝑛 plus one is equal to one over 𝑛 plus one squared plus one, which after we distribute the parentheses, we can see is equal to one over 𝑛 squared plus two 𝑛 plus two. Now, we’re ready to find our value of 𝐿. We have that 𝐿 is equal to the limit as 𝑛 tends to ∞ of π‘Ž 𝑛 plus one over π‘Ž 𝑛. Substituting in our values for π‘Ž 𝑛 and π‘Ž 𝑛 plus one, we can see that this limit is equal to the limit as 𝑛 tends to ∞ of the absolute value of 𝑛 squared plus one over 𝑛 squared plus two 𝑛 plus two.

Now, we can divide both the numerator and denominator of the fraction here by 𝑛 squared. And we’ll do this because we’re trying to find the infinite limit of a rational function. And 𝑛 squared is the highest power which occurs in our fraction. So we have the limit as 𝑛 tends to ∞ of the absolute value of one plus one over 𝑛 squared all over one plus two over 𝑛 plus two over 𝑛 squared. Next, we used the fact that the limit as 𝑛 tends to ∞ of one over 𝑛 is equal to zero. Therefore, any term within our limit, which uses one over 𝑛 or one over 𝑛 squared, will tend to zero as 𝑛 tends to ∞.

And so our limit will become the absolute value of one plus zero over one plus zero plus zero. And this is simply equal to one. Therefore, we found that 𝐿 is equal to one. Looking at the ratio test, we can see that this satisfies condition number three. Therefore, we can say that the ratio test is inconclusive. It doesn’t tell us whether this series is absolutely convergent, conditionally convergent, or divergent. Therefore, the answer to this question is false.

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