I want to re-arrange the terms of the alternating harmonic series so that they'll converge to 1.5 instead of being between .5 and 1.

(This can be done for any positive number)

1) Add up enough positive terms to just go over 1.5 (in order of their appearance)

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2) Include however many negative terms I need to in order to go below 1.5 (in order of their appearance)

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3) Now include the next positive terms, until I just go over 1.5 again.

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4) You get the picture: include the next negative terms, until I just go under 1.5 again.

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5) Include the next positive terms, until I just go over 1.5:

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6) Include the next negative terms, until I just go under 1.5:

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7) Now for the positive terms again:

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While it may seem like we'll exhaust the positive terms before the negative terms, because we're using five positive terms for every one negative term ... there's an infinite supply of positive terms, so they can't be exhausted.

Let's go a bit further

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Do you have doubts?

Look at the terms of the sequence of these re-arranged partial sums:

1.533333333

1.033333333

1.521800422

1.271800422

1.514352839

1.347686172

1.510338491

1.385338491

1.407874824

1.506217169

1.422883836

1.505027922

1.433599351

1.504133948

1.441633948

1.503437767

1.447882211

1.502880434

1.452880434

If you saw these terms without knowing where they came from, you'd have little doubt that it's converging to something between 1.453 and 1.503. I can continue to take this out however far I want to, and I just keep getting closer and closer to 1.5!

Bizarre, huh?

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