Determine whether the sequence (-1)^n(n/(n+1)) diverges or converges.

I think it diverges because the limit as n approaches inf. of n/(n+1) is 1, and the limit of (-1)^n does not exist.

If you mean this:

lim ((-1)^n )(n/(n+1) correct.

To determine whether the sequence (-1)^n(n/(n+1)) converges or diverges, we need to analyze the behavior of each component of the sequence.

First, let's consider the sequence (n/(n+1)). As n approaches infinity, the expression n/(n+1) approaches 1. This can be seen by dividing every term in the sequence by n and taking the limit:

lim(n->inf) (n/(n+1)) = lim(n->inf) (1/(1 + 1/n)) = 1/(1 + 0) = 1

So, the sequence (n/(n+1)) converges to 1 as n approaches infinity.

Now, let's consider the sequence (-1)^n. This is an alternating sequence that oscillates between -1 and 1 as n increases. It does not approach a specific value as n approaches infinity.

Combining the two sequences, we have (-1)^n(n/(n+1)). Since one component converges to 1 and the other oscillates, the entire sequence does not converge to a single value. Therefore, the sequence (-1)^n(n/(n+1)) diverges.

To determine whether the sequence (-1)^n(n/(n+1)) converges or diverges, we need to consider the behavior of each factor individually.

The first factor, (-1)^n, alternates between being -1 when raised to an odd power (n = 1, 3, 5, ...) and 1 when raised to an even power (n = 2, 4, 6, ...). Since this factor does not approach a specific value but rather oscillates between -1 and 1, its limit does not exist.

The second factor, n/(n+1), approaches 1 as n approaches infinity:
lim (n/(n+1)) = 1

Now, to determine the overall behavior of the sequence, we multiply the two factors together:
(-1)^n * (n/(n+1)) = (-1)^n(n/(n+1))

Since the limit of (-1)^n does not exist and the limit of (n/(n+1)) is 1, we can conclude that the given sequence diverges because it does not approach a specific value.

Therefore, (-1)^n(n/(n+1)) is a divergent sequence.