Determine if the following series converges or diverges by using limit comparison test.

Solution

To make use of the limit comparison test we require to find out a second series that we can find out the convergence of easily and has what we assume is similar convergence as the specified series.  On top of that we will need to select the new series in such type of a way as to give us an easy limit to calculate for c.

We have already guessed that this series converges and as it's vaguely geometric let us use

as the second series.  We make out that this series converges and there is a chance that as both series has the 3n in it the limit won't be too bad.

Here's the limit.

Now here, we'll require to make use of L'Hospital's Rule on the second term to in fact evaluate this limit.

Thus, c is positive and finite so by the Comparison Test both series should converge since

converges.