Beck said "show us what you showed me in my office" implying that he'd been shown the problem before.
But it should be fairly obvious to any calculus student that that series doesn't converge since the limit as it goes to infinity isn't 0... It doesn't even exist. Sine and cosine functions are an immediate red flag to this, all you need to do to look at it is a simple order of magnitude analysis.
Nevertheless, I'm sure he's very smart. But I'm not sure he has much common sense if he didn't see that right away. I'll also echo Alex's sentiments below...
I'm friends with an incredibly smart kid. He's 14, but has been put up three grades in school at one point. He does all the obvious enrichment things which are available in the relatively small Australian city he lives in.
His life experience has been pretty unusual. He doesn't really know what it's like to be challenged in school. All his friends are way older than he is. (Once, I asked him how being constantly around people older than him made him feel. He replied, "Concerned for my future.")
He doesn't know anyone like him, which I think is a shame: he'd probably get along very well with them.
Does anyone know any similar kid geniuses? If so, can I give them my friend's details?
Thanks.