4.1. Series and Convergence

Examples 4.1.7(a): Rearranging the Alternating Harmonic Series

Find a rearrangement of the alternating harmonic series that is within 0.001 of 2, i.e. show a concrete rearrangement of that series that is about to converge to the number 2.
This is a simple numeric computation, involving some trial and error. First, let's collect positive odd terms so that they add up to something larger than 2:
1 + 1/3 + ... + 1/15 = 2.021800422
Next we subtract the first negative term:
1 + 1/3 + ... + 1/15
      - 1/2
      = 1.521800422
Then we again add positive terms until we are larger than 2 again:
1 + 1/3 + ... + 1/15
      - 1/2
      + 1/17 + 1/19 + ... + 1/41
      = 2.004063454
Subtracting the next negative term gives:
1 + 1/3 + ... + 1/15
      - 1/2
      + 1/17 + 1/19 + ... + 1/41
      - 1/4
      = 1.754063454
Again adding positive terms:
1 + 1/3 + ... + 1/15
      - 1/2
      + 1/17 + 1/19 + ... + 1/41
      - 1/4
      + 1/43 + 1/45 + ... + 1/69
      = 2.009446048
Once again we subtract the next negative term:
1 + 1/3 + ... + 1/15
      - 1/2
      + 1/17 + 1/19 + ... + 1/41
      - 1/4
      + 1/43 + 1/45 + ... + 1/69
      - 1/6
      = 1.842779381
and add positive terms:
1 + 1/3 + ... + 1/15
      - 1/2
      + 1/17 + 1/19 + ... + 1/41
      - 1/4
      + 1/43 + 1/45 + ... + 1/69
      - 1/6
      + 1/71 + 1/73 + ... + 1/95
      = 2.000697893
Now we are within 0.001 of 2, and it seems clear how we can continue in this matter to find a rearrangement of the alternating harmonic series that converges to 2.
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