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# Does anyone want a really enjoyable Maths problem I've been thinking about for days? Answered

Here is it:
Prove that (Xn): (any) n Xn=1+1/1!+1/2!+1/3!.....+1/n! converges and lim(Xn)=lim(Yn), where
n
Yn=(1+1/n)

( (any) is upside down A, n in the end is power)
I know asking homework-based questions is unethical, but I've been doing it for 2 weeks and my teacher tells me it is very enjoyable...

Any ideas?

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## 4 Replies

kelseymh (author)2010-10-13

I'm not a good enough mathematician to see why the Bolzano-Weierstrass existence theorem applies in this particular case.

All I could do is to churn through the math; in particular doing the expansion of Yn and showing that it's the same as the sum of inverse factorials term by term. Brute force is never elegant :-(

Asking homework-based problems is frowned upon around here only if you don't disclose it. You haven't done that, and you've given the background that this isn't just "ooh, I need an answer for tomorrow's exam." :-) I for one am interested to see what our mathematician members come up with...