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

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?

## Discussions

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 Y

_{n}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...

Sometimes you guys frighten me.

The problem was almost right after the Bolzano-Weierstrass theorem in the paper, and as Xn+1=Xn + 1/(n+1)! it is monotonous and rising, and if I prove that (exist) C: (any) n Xn0 than it converges, and sup Xn =lim Xn. Maybe it can help...

Tried an inductive method ?