Computation shows that
is, for "large n" (10 or greater, say), just a little less than an integer. For example,
Why is this? (I do not know the answer.)
A related thread is here:
http://www.mathhelpforum.com/math-he...r-divisib.html
Computation shows that
is, for "large n" (10 or greater, say), just a little less than an integer. For example,
Why is this? (I do not know the answer.)
A related thread is here:
http://www.mathhelpforum.com/math-he...r-divisib.html
If the integer part of x is denoted by [tex][x][/tex] fractional part of x is denoted by [LaTeX ERROR: Compile failed] , then you can see the following:
[LaTeX ERROR: Compile failed]
[LaTeX ERROR: Compile failed]
[LaTeX ERROR: Compile failed]
[LaTeX ERROR: Compile failed]
[LaTeX ERROR: Compile failed]
You can formulate it as a statement: f(n) gets arbitrarily close to an integer as n goes to infinity. Then maybe you can prove it. This is better than what you state, because we not only see that the function is close to an integer for big n, but it's getting closer and closer to an integer.
[LaTeX ERROR: Compile failed]
Here's an outline of a proof. No time for details because I'm about to go away for a week's holiday (again — that's the advantage of being retired).
The number satisfies the equation (see here). The other two roots of this equation, call them and , satisfy and .
Define a sequence of integers by and for .
Then the theory of difference equations shows that . Since and as n gets large, it follows that is only slightly different from an integer.