Granted, they are playing fast and loose, but "evaluating" any infinite series (especially those that aren't absolutely convergent) requires some artifice. But mathematics is full of artifices -- it's just that some are much more familiar than others (e.g., complex numbers, or even negative numbers).
The mathematically-educated tend to fetishize mathematical rigor, especially as they learned it (e.g., epsilon-delta arguments). This is not only extremely ahistorical but also leads to a banal view of mathematics (and I'd say that undergraduate mathematics is nearly entirely banal). When presented with some curious but dubious claims, the response should be "what's going on?" not "this is bullshit".