122Views4Replies

# 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?

## Comments

10 years ago

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...

Answer 10 years ago

Sometimes you guys frighten me.

Answer 10 years ago

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...

10 years ago

Tried an inductive method ?