Well, plugging in infinity as you've done there isn't really allowed as a rigorous step, but it's alright to get an intuition of the long-term behavior of the series. Essentially, what you've done thus far is found that \(\displaystyle \displaystyle lim_{n \to \infty} \: U(n) = 0\). In other words, as
n grows without bounds, the individual terms of both series approach 0. I don't know what your textbook/class calls the various theorems, but when I was learning about series, my book had something called
The Divergence Test. It says that if the limit of the terms of a series
doesn't go to 0, the series must diverge. However, if the limit of the terms does go to 0, we cannot say one way or the other if a series converges or diverges. Right now, you've found that the divergence test is inconclusive. When one convergence test comes back inconclusive, that means you have to try another one.
What you're talking about at the end seems to be another test for the convergence of a series. My book called that one the P-Test. Unfortunately, it cannot be used here, because neither of the series are in the proper form. As you note the series must have the form 1/n^p, where n is the index variable and p is any constant. That's not the case though. In each of the series the variable
n is in the exponent, not the base.
However, there's still hope. If we rewrite the first series a bit, we can see that U(n) = -1/3^(2n-1) = -(1/3)^(2n-1). That looks like a geometric series. Is there perhaps a convergence test you know about geometric series? (If not, you might try
this page which has a bunch of convergence tests). What does that test tell you about whether the first series converges or diverges? Similarly, the second series can also be written as a geometric series. I'll leave you to see if you can find what it is. Then, using the same test, does it converge or diverge? Now, given that the main series the problem asks about can be written as the sum of the two individual series, what do these two results tell you?
