Why is the limit of the sum n=1 to infinity of ln(2(n+1))–ln(2n) divergent? I thought it would be -ln2 because all the other terms cancel out cancel out.

Answer 1

L'Hopitals rule:

(1/2n+2)(2n/1)

2n/(2n+2)

second application of the rule:
2/2=1

so at infinity, you are adding 1 + 1+ ...

which means it is divergent.