Homework Section 11.2, #16, 20, 44, pp. 735--

#16: 
a) there is no difference between the two -- except for the name of the dummy index.
b) The second series is equal to n*a_j, whereas the first is the sum of the first n terms of the sequence.

#20: 2 + 0.5 + 0.125 + 0.03125 + ...

c=2
r=.5/2
S=c/(1-r)

#44: divergent, as one can see using properties of ln:

a[n_]:=Log[n/(n+1)]
b[n_]:=Log[n]-Log[n+1]
a[Range[10]]
b[Range[10]]
Simplify[Total[a[Range[10]]]]