So if you take an alternating series and miss out lots of terms then the partial sum is wrong. How is that bizarre or even interesting? By rearranging the terms and 'forgetting' a few in the new "partial sum" you have made a mathematical error. Obviously if you just rearranged ALL the terms (up to the Nth in the original series) the partial sum result would be the same.

Imagine you put all the positive terms first, letting the negative ones fall off the end because you were only looking at a partial sum. That would not be a mathematically sound approach. What you have shown is something approaching that scheme, but not quite as bad. I imagine that what you have presented is "orthodox" maths. Nevetheless it is a work of sleight of hand, and not sensible. Just because it is an infinite series does not allow one to go crazy!