Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
An=[(-1)^n]/[8[(n)^(1/2)]]
since the terms are just basically 1/√n the sequence converges to 0
To determine if a sequence converges or diverges, we need to find its limit as n approaches infinity.
The given sequence is: An = ((-1)^n) / (8√n)
Let's first consider the numerator, (-1)^n. When n is an even number, (-1)^n gives us 1, and when n is an odd number, (-1)^n gives us -1. This means that the numerator alternates between 1 and -1 as n increases.
Next, let's consider the denominator, 8√n. As n approaches infinity, the square root of n also approaches infinity.
Putting it all together, we have:
An = (1 or -1) / (8√n)
As n approaches infinity, the value of √n becomes very large. This means that the denominator, 8√n, approaches infinity as well.
Now let's analyze the numerator. Since it alternates between 1 and -1, as n increases, the numerator does not approach a specific value. It oscillates between 1 and -1 indefinitely.
Therefore, when you divide 1 or -1 by a value that approaches infinity, the result does not converge to a single value. Hence, the given sequence diverges.
The answer is: The sequence diverges.