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I saw a demonstration of this sum recently. It went like this:
S=11+11+...
=1(11+1...)
the periods indicating a continuation to infinity, it is assumed that:
S=1S
2S=1
S=1/2
Then again, the alternating harmonic series converges to ln2 but by rearranging the terms in the equation, it also converges to (ln2)/2 and (ln2)/4 and so forth. The problem in your reasoning is that you take a partial sum as reference. I know that it's counterintuitive just like 0.00000...1= 0 but the reason it gives that result is only because the sum is infinite. The moment it becomes finite, these results all go wrong.