For the ratio test of a series the ratio is defined as:
And we have three possible outcomes:
- r<1 and the series converges.
- r>1 and the series diverges.
- r=1 and the test is inconclusive.
Knowing this let's apply the test to the series given by the question. We have:
As you can see both expressions are always positive so when writing the ratio we don't need to add the absolute value symbols. Then the ratio is:
Here is important to remember a property of the factorial:
And a property of powers:
Using these properties we get:
So the answer to part (a) is:
For part (b) we have:
The ratio depends on n. As n increases the ratio increases and gets greater than 1:
So for any n>3 the ratio is greater than 1 which means that this value of r tells us that the series diverges.