Calculus II  
Quiz #10
May 8, 2002
Name____________________
R.  Hammack 
Score ______


(1) Decide if the following series converge or diverge. Explain your reasoning completely.  


(a)   [Graphics:Images/quiz11sol_gr_1.gif]

This is an alternating series  [Graphics:Images/quiz11sol_gr_2.gif] with  [Graphics:Images/quiz11sol_gr_3.gif] and [Graphics:Images/quiz11sol_gr_4.gif].
Therefore, by the Alternating Series Test, it converges.


(b)  [Graphics:Images/quiz11sol_gr_5.gif]
Notice that     [Graphics:Images/quiz11sol_gr_6.gif],   and    [Graphics:Images/quiz11sol_gr_7.gif] is a divergent p-series.
Therefore, by the comparison test,   [Graphics:Images/quiz11sol_gr_8.gif] diverges.



(c)   [Graphics:Images/quiz11sol_gr_9.gif]
Using the Ratio Test,   [Graphics:Images/quiz11sol_gr_10.gif][Graphics:Images/quiz11sol_gr_11.gif][Graphics:Images/quiz11sol_gr_12.gif].
It follows that the given series converges.



(d)  
[Graphics:Images/quiz11sol_gr_13.gif]
This series has positive and negative terms, but it's not alternating. Therefore, we check for absolute convergence. That involves investigating the series [Graphics:Images/quiz11sol_gr_14.gif]. Notice that [Graphics:Images/quiz11sol_gr_15.gif]< [Graphics:Images/quiz11sol_gr_16.gif]and it follows that [Graphics:Images/quiz11sol_gr_17.gif] converges by comparison with the convergent geometric series [Graphics:Images/quiz11sol_gr_18.gif][Graphics:Images/quiz11sol_gr_19.gif]. Consequently, the original series converges absolutely, so it converges.



(e)
[Graphics:Images/quiz11sol_gr_20.gif]

[Graphics:Images/quiz11sol_gr_21.gif][Graphics:Images/quiz11sol_gr_22.gif][Graphics:Images/quiz11sol_gr_23.gif][Graphics:Images/quiz11sol_gr_24.gif][Graphics:Images/quiz11sol_gr_25.gif][Graphics:Images/quiz11sol_gr_26.gif][Graphics:Images/quiz11sol_gr_27.gif] Therefore, by the Divergence Test, the series diverges.