Date: Feb 15, 2014 5:25 AM
Author: Karl-Olav Nyberg
Subject: Re: on real part of [(1+isqrt(7))/2]^n

On Saturday, February 15, 2014 10:29:28 AM UTC+1, quasi wrote:
> >konyberg
>
> >
>
> >|a_n| = |(sqrt(2))^n * cos(n*acos(sqrt(2)/4))|
>
> >
>
> > = (sqrt(2))^n * |cos(n*acos(sqrt(2)/4))|
>
> >
>
> >where |cos(n*acos(sqrt(2)/4))| can be considered as a "constant"
>
> >different from 0.
>
>
>
> Certainly not!
>
>
>
> Different from 0 doesn't mean it can be considered as a "constant".
>
>
>
> For example, how do you know that there aren't infinitely positive
>
> integers n such that
>
>
>
> |cos(n*acos(sqrt(2)/4))| < 1/sqrt(2)^n
>
>
>
> ??
>
>
>
> You don't.
>
>
>

> >Conclusion: The limit |a_n| goes to infinity as n goes to
>
> >infinity.
>
>
>
> No, it's not even close to a proof.
>
>
>
> quasi


Hi.

You are as always correct! My initial stand that the limit is undefined was based on your argument.
However: I have done some probing with |a_n| using Excel. The point function seems to have a tendence to go up. Not monotonally, but still. I think that the probality that the limit is infinity is high, but not proven to be infinity.

This problem is actually fun to investigate!

Your post with the conjucture and assumed proof was great. Thank You.

KON