Using only the ratio test, determine whether or not the recursive sequence
converges or diverges.
Applying the ratio test yields
Since the limit equals , the ratio test tells us that the series converges.
Explanation of Each Step
To apply the ratio test, we must evaluate the limit
In our problem, we can use
and substitute this into our limit.
In Step (2), we only cancel the in the numerator and denominator.
First observe that
Dividing everything by the square root of we obtain
In Step (4) we only evaluate the limit:
which equals zero because the numerator is a constant and the denominator goes to infinity.
In Step (5) we apply the Squeeze Theorem.
Potential Challenge Areas
Because the question asks us to apply the ratio test, we know that we will start our solution by using the formula
Most problems involving convergence tests don't involve recursive formulas. But with the ratio test, we apply
and use the given recursion equation for . In our case, our recursion equation is
which we substitute into the numerator, allowing us to cancel the in the numerator and denominator. This trick is a bit harder to apply for the other convergence tests.