Continuous Fraction Series

[rheighton]

I have a series as follows:

(sorry for terrible editing on this one... it basically is = 1 + 1/(1+...1+1/1))

1
An= 1 + ----------------------------
1 + 1
----------------------
1 + 1
---------------
1+...+1+1/1

where n>=0 represents the number of fraction lines in the expression of An

I am asked to do the following:
1) find A0, A1, A2 ----> was able to do this with no problem

2) Use induction to show that An>=1 for all n>=0 ----> do I just repeat terms of A, making n larger and larger, thus proving that it just becomes slightly larger fractions?

3) Prove that An is convergent ----> which I think is done in 4) and 5):

4) Develop a recursive formula to generate An ----> I worked this out to be An = 1+ (n-1)/n but I may be wrong

and finally
5) Find the limit of An as n approaches infinity ----> do I assume An to be the recursive formula worked out in 4)? If this is true, I worked the limit out to be 2, as I have 1 + [infinity/infinity (equals 1?)]

And am I proving the convergence by finding the limit of An?

Thanks so much for any help, I've been racking my brain on this one!

 

[jolly]

I don't think your "problem" came out right.

 

[rheighton]

Yes, I edited the post, and summed the problem up more simply in the second line as An = 1 + 1/(1+...1+1/1)), with n being the number of fraction lines

 

[royhaas]

Are you talking about \(\displaystyle A_{n+1} = 1+1/A_n\), perhaps?

 

[rheighton]

Yup, that seems to be my problem... I'm pretty sure that my recursive formula for An is right...