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An extended harmony (Posted on 2011-11-01) Difficulty: 3 of 5
How many members of the harmonic series 1+1/2+1/3+1/4+ +1/n are needed to add up close to 10 , without going over it?

No Solution Yet Submitted by Ady TZIDON    
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A simple upper bound | Comment 1 of 5
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
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