(1) Decide if the following series converge or diverge, and explain your reasoning.  If a series converges, say what it equals, if possible.


(a)   12/5
(Convergent geometric series with a=3 and r=-1/4.)

(b)    This is a p-series with p=1/3. It diverges because 0<p<1.

(c)   
This series is the sum of a convergent geometric series (a=1, r=1/2) and a convergent p-series (p=2>1).
Therefore the series converges, but unfortunately we can't say what it converges to because there is no formula for the value of the p-series.

(d)  
This is neither a geometric series nor a p-series.

Applying the divergence test, we get , so there is no conclusion.

Since we couldn't rule out divergence, let's see if the integral test applies. Let .  In order for the integral test to apply, this function must have positive terms and decrease on some interval [a, ∞). All terms are clearly positive (except the first, which is 0). Also, as = , we see that    is negative for all x >, and therefore f decreases on [). To simplify our work, we can be assured that f decreases on [2,∞). Therefore the integral test applies, and we need to investigate

.
Since the integral converges, the integral test implies that the series converges as well.