Determine if the subsequent series is convergent or divergent.

Solution

As the cosine term in the denominator doesn't get too large we can suppose that the series terms will behave such as,

n / n² = 1/n

which, as a series, will diverge.  Thus, from this we can guess that the series will possibly diverge and thus we'll need to find out a smaller series that will as well diverge.

 Remind that from the comparison test with improper integrals that we ascertained that we can make a fraction smaller by either creating the numerator smaller or the denominator larger.  In this example the two terms in the denominator are both positive. Thus, if we drop the cosine term we will actually be making the denominator larger as we will no longer be subtracting off a positive quantity.  Hence,

n/ (n² - cos² (n)) > n/n² = 1/n

Then, as

diverges (it's harmonic or the p-series test) by the Comparison Test our original series must as well diverge.