Question Video: Using the Ratio Test to Determine Convergence Mathematics • Higher Education

Video Transcript

Consider the series the sum from 𝑛 equals zero to ∞ of 𝑎 𝑛, where 𝑎 𝑛 is equal to 𝑛 plus 𝑏 factorial divided by 𝑐 to the 𝑛th power for some integers 𝑏 and 𝑐, both strictly greater than one. Calculate the limit as 𝑛 approaches ∞ of the absolute value of 𝑎 𝑛 plus one divided by 𝑎 𝑛. Hence, decide whether the series converges or diverges.

The question gives us a series, the infinite sum of 𝑎 𝑛, where it tells us that 𝑎 𝑛 is equal to 𝑛 plus 𝑏 factorial all divided by 𝑐 to the 𝑛th power, where 𝑏 and 𝑐 are integers which are strictly greater than one. The first part of this question wants us to calculate the limit as 𝑛 approaches ∞ of the absolute value of the ratio of successive terms in the sequence 𝑎 𝑛. The first thing we notice is, instead of dividing by 𝑎 𝑛 in our limit, we can multiply by the reciprocal of 𝑎 𝑛. So we want to calculate the limit as 𝑛 approaches ∞ of the absolute value of 𝑎 𝑛 plus one multiplied by the reciprocal of 𝑎 𝑛.

To calculate these values, we’ll use the definition of 𝑎 𝑛 given to us in the question. We have that 𝑎 𝑛 plus one is equal to 𝑛 plus one plus 𝑏 factorial all divided by 𝑐 to the power of 𝑛 plus one. And we multiply this by the reciprocal of 𝑎 𝑛, which is 𝑛 plus 𝑏 factorial divided by 𝑐 to the 𝑛th power. We then take the reciprocal of 𝑎 𝑛. This gives us the limit as 𝑛 approaches ∞ of the absolute value of 𝑛 plus one plus 𝑏 factorial divided by 𝑐 to the power of 𝑛 plus one multiplied by 𝑐 to the 𝑛th power divided by 𝑛 plus 𝑏 factorial. We’re now ready to start simplifying.

First, we notice that 𝑛 plus one plus 𝑏 factorial is actually equal to 𝑛 plus one plus 𝑏 multiplied by 𝑛 plus 𝑏 factorial. This means we can cancel the shared factor of 𝑛 plus 𝑏 factorial in our numerator and our denominator. Similarly, we can cancel 𝑛 of the shared factors of 𝑐 in our numerator and our denominator. This gives us the limit as 𝑛 approaches ∞ of the absolute value of 𝑛 plus one plus 𝑏 all divided by 𝑐. We see that one does not vary depending on the value of 𝑛. We also see that 𝑏 does not vary depending on the value of 𝑛. And we also see that 𝑐 does not vary depending on the value of 𝑛. However, our term of 𝑛 is getting larger and larger as 𝑛 is approaching ∞. Therefore, the numerator of our fraction is approaching ∞. However, the denominator remains constant. Therefore, we can evaluate this limit as ∞.

Finally, the question wants us to use this to determine whether the series converges or diverges. To do this, we recall the ratio test, which tells us if the limit as 𝑛 approaches ∞ of the absolute value of the ratio of successive terms is less than one. Then the sum from 𝑛 equals zero to ∞ of 𝑎 𝑛 converges absolutely. And if the limit as 𝑛 approaches ∞ of the absolute value of the ratio of successive terms is greater than one. Then the sum from 𝑛 equals zero to ∞ of 𝑎 𝑛 diverges. And we already calculated the limit as 𝑛 approaches ∞ of the absolute value of the ratio of successive terms. And we showed it to be ∞. And in the case of the ratio test, this also counts as this limit being greater than one. Therefore, we can conclude that our series must diverge.

Therefore, for the sequence 𝑎 𝑛 is equal to 𝑛 plus 𝑏 factorial all divided by 𝑐 to the 𝑛th power for some integers 𝑏 and 𝑐 both greater than one. Since the absolute value of the ratio of successive terms is equal to ∞, we can conclude by the ratio test that the series must diverge.