At the end of class, one student asked whether 
assumption (a) that b_{n+1} <= b_n in the 
Alternating Series Test was necessary. 
He wanted to know if lim b_n=0 is enough
and he couldn't see why part a) was needed.

Here is an example of an Alternating Series whose n-th term is
(-1)^{n-1} b_n that satisfies b_n ---> 0 as n --> infinity but
which does NOT converge because it fails assumption (a):

b_1 = 1 + (1/2)

b_2 = 1

b_3 = 1 + (1/2)

b_4 = 1   (note: The partial sum S_4 = 1)

b_5 = (1/2) + (1/3)

b_6 = (1/2)

b_7 = (1/2) + (1/3)

b_8 = (1/2)

b_9 = (1/2) + (1/3)

b_10 = (1/2)   (Note: The partial sum S_10 = 2)

b_11 = (1/3) + (1/4)

b_12 = (1/3)

b_13 = (1/3) + (1/4)

b_14 = (1/3)

b_15 = (1/3) + (1/4)

b_16 = (1/3)

b_17 = (1/3) + (1/4)

b_18 = (1/3)  (Note: The partial sum S_18 = 3)

b_19 = (1/4) + (1/5)

b_20 = (1/4), etc...