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Theorem: 1 = 1/2 [For "Series"-ous Minded Students]

Proof:

Start with the infinite series 1/(1*3) + 1/(3*5) + 1/(5*7) + 1/(7*9) + ...

Re-write it as 1/2[(1/1 - 1/3) + (1/3 - 1/5) + (1/5 - 1/7) + (1/7 - 1/9) + ... ]

Go back and check that the above series are the same!

Since all terms after 1/1 cancel, the sum of the series is 1/2.

Now, we can also re-write the original series as (1/1 - 2/3) + (2/3 - 3/5) + (3/5 - 4/7) + (4/7 - 5/9) + ...

Again, all terms after 1/1 cancel, so that the sum is 1.

Thus 1/2 = 1.

What's wrong?