Question Video: Using the Ratio Test for Series Convergence Mathematics • Higher Education

Video Transcript

Using the ratio test, determine whether the series the sum from 𝑛 equals one to ∞ of three to the power of 𝑛 over two 𝑛 factorial is convergent, divergent, or whether the test is inconclusive.

We’re told to use the ratio test. Now, given a series the sum of 𝑎 𝑛, we can calculate the limit as 𝑛 approaches ∞ of the absolute value of 𝑎 𝑛 plus one over 𝑎 𝑛. Let’s say we get some constant 𝐿. If 𝐿 is less than one, then that tells us that the series the sum of 𝑎 𝑛 is absolutely convergent. If it’s greater than one or in the ∞, then the series is divergent. And if it’s equal to one, the ratio test is inconclusive. That is, no conclusion can be drawn about the convergence or divergence of the sum of 𝑎 𝑛.

If we look at our series, we see that 𝑎 𝑛 is equal to three to the power of 𝑛 over two 𝑛 factorial. 𝑎 sub 𝑛 plus one is found by replacing 𝑛 with 𝑛 plus one. And we get three to the power of 𝑛 plus one over two times 𝑛 plus one factorial. And in fact, if we distribute the parentheses on our denominator, we get 𝑎 sub 𝑛 plus one to be equal to three to the power of 𝑛 plus one over two 𝑛 plus two factorial. In the ratio test, we need to find the limit as 𝑛 approaches ∞ of the absolute value of the quotient of these. So that’s three to the power of 𝑛 plus one over two 𝑛 plus two factorial divided by three to the power of 𝑛 over two 𝑛 factorial.

Now, we know that dividing by a fraction is the same as multiplying by the reciprocal of that fraction. So our limit is three to the power of 𝑛 plus one over two 𝑛 plus two factorial times two 𝑛 factorial over three to the power of 𝑛. When we divide two numbers whose base is equal, we simply subtract their exponents. So three to the power of 𝑛 plus one divided by three to the power of 𝑛 is three to the power of 𝑛 plus one minus 𝑛, which is three to the power of one or simply three.

Next, we recall that two 𝑛 plus two factorial is two 𝑛 plus two times two 𝑛 plus one times two 𝑛 and so on or two 𝑛 plus two times two 𝑛 plus one times two 𝑛 factorial. So when we have two 𝑛 factorial over two 𝑛 plus two factorial, the two 𝑛 factorials cancel. So we’re left with the limit as 𝑛 approaches ∞ of the absolute value of three over two 𝑛 plus two times two 𝑛 plus one. And we can, if we wish, take out a constant factor of three. So we have three times the limit as 𝑛 approaches ∞ of one over two 𝑛 plus two times two 𝑛 plus one.

Now, as 𝑛 gets very, very large, the denominator of the fraction one over two 𝑛 plus two times two 𝑛 plus one also gets very, very large. And this means one over two 𝑛 plus two times two 𝑛 plus one itself gets very, very small. It approaches zero. And so, the limit as 𝑛 approaches ∞ of the absolute value of one over two 𝑛 plus two times two 𝑛 plus one is zero. So we have three times zero, which is equal to zero. And so, since our limit is zero, it’s less than one. We can see that our series must be convergent. And so, we’re done. Using the ratio test, we found that the series the sum from 𝑛 equals one to ∞ of three to the power of 𝑛 over two 𝑛 factorial is convergent.