Alternating series test

Question

determine whether the series below converges, absolutely converges, conditionally converges or diverges. Make sure to identify which test is being used.

The series = [sum sign] n=1 to infinity [(-1)ⁿ⁺¹(n+2)]/[n(n+3)]

My approach to the question - using the alternating series i was able to identify that the series converges because it keeps alternating between positive and negative, keeps decreasing and if we take the limit of the function we can see that it goes to 0.

Now how can I find out whether it converges, absolutely converges, conditionally converges or diverges. What steps can be used to identify it. I was thinking limit comparison test but what would we compare the series too and what are the steps that can lead us to whether it converges, absolutely converges, conditionally converges or diverges.

Another test that comes to mind is the integral test which says that the series diverges so then we can conclude the series conditionally converge.

Answer 1

Step 1: Test for convergence using the Alternating Series Test

We start by recognizing that this is an alternating series, since it includes the term (-1)^{n+1}. To apply the Alternating Series Test, we need to check three things:

1. The terms without the sign alternation are positive. That is, a_n=(n+2)/(n(n+3))>0
2. The sequence a_n is decreasing. Although we could use calculus to show this, we can also observe that as n increases, the numerator grows slowly while the denominator grows faster, so a_n is eventually decreasing.
3. The limit of a_n as n → infinity is zero. This is true because the denominator, which is of degree 2 (quadratic), grows faster than the numerator, which is of degree 1 (linear).

Since all three conditions are satisfied, the series converges by the Alternating Series Test.

Step 2: Test for Absolute Convergence

Now consider the series without the alternating sign:

Sum from n = 1 to infinity of |(-1)^(n+1)*(n+2)/(n(n+3))|

We'll compare this to the harmonic series Sum from n = 1 to infinity {1/n}, which diverges.

By the Limit Comparison Test:

limit as n approaches infinity {(n+2)/(n(n+3))} divided by {1/n}

Since the limit is a positive finite number and Sum {1/n } diverges, it follows that:

Sum from n = 1 to infinity of (n+2)/(n(n+3))

also diverges.

Therefore, since the original alternating series converges and the absolute value series diverges, the original series converges conditionally (not absolutely).