Calculus II 
Quiz #9
April 26, 2002
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)   [Graphics:Images/quiz9sol_gr_1.gif][Graphics:Images/quiz9sol_gr_2.gif][Graphics:Images/quiz9sol_gr_3.gif][Graphics:Images/quiz9sol_gr_4.gif]12/5
(Convergent geometric series with a=3 and r=-1/4.)

(b)    [Graphics:Images/quiz9sol_gr_5.gif][Graphics:Images/quiz9sol_gr_6.gif] This is a p-series with p=1/3. It diverges because 0<p<1.

(c)    [Graphics:Images/quiz9sol_gr_7.gif] [Graphics:Images/quiz9sol_gr_8.gif]
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.

This is neither a geometric series nor a p-series.

Applying the divergence test, we get [Graphics:Images/quiz9sol_gr_10.gif], so there is no conclusion.

Since we couldn't rule out divergence, let's see if the integral test applies. Let [Graphics:Images/quiz9sol_gr_11.gif].  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 [Graphics:Images/quiz9sol_gr_12.gif]= [Graphics:Images/quiz9sol_gr_13.gif], we see that    [Graphics:Images/quiz9sol_gr_14.gif]is negative for all x >[Graphics:Images/quiz9sol_gr_15.gif], and therefore f decreases on [[Graphics:Images/quiz9sol_gr_16.gif]). 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.