Writing a program would not be much of a challenge even for me so I thought Id see what I can do with p&p.
The classic proof that the harmonic series diverges is useful.
round each term whose denominator is not a power of 2 down so that it is. 1/3 becomes 1/4. 1/5, 1/6, and 1/7 become 1/8 etc. This creates a smaller sequence.
Now there are two 1/4s which add to 1/2, there are four 1/8s which add to 1/2 etc.
Since the first term is 1, you will need 18 of these 1/2s, the last one being from a bunch of 1/2^18
So if you have 2^18 = 262144 terms it will definitely be over 10.
Posted by Jer
on 2011-11-01 10:19:57