Calculus II 
Quiz #9

April 26, 2002

Name____________________ 
R. Hammack

Score ______

(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 pseries 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
pseries (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 pseries.
(d)
This is neither a geometric series nor a pseries.
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.