Determine if the series converges or diverges.

Answer

The divergence test is no help. The limit of the terms is zero:

This means there's hope for the series to converge and we need to try another test. This isn't a geometric series or an alternating series, it doesn't look like an integrable function, and it doesn't look very ratio-like (no exponents or factorials). That leaves the comparison test.

Since the denominator has a 3^{n} we guess the series converges. We need to find a series with bigger terms that also converges. Thankfully, since making the denominator smaller makes the fraction bigger,

(we should point out that everything in sight is positive, so the required condition that the terms be non-negative is met). The series

converges (it's a geometric series with |*r*| < 1, or you could use the ratio test if you really wanted). This means the smaller series

also converges.