What is the 1000th Fibonacci number divided by the 1001 Fibonacci number?

The 960^th Fibonacci number is already larger than (1 googol)² , so a precise,
unrounded answer to your question would have more than 10²⁰⁰ digits in it,
and you're not going to find anybody outside of a college supercomputer lab
who can work with the numbers you're asking about.

For the same reason, we can be pretty sure that you have no earthly use for
the answer either, so we're not particularly bothered by the fact that we can't
provide it.

However, don't despair. All is not lost! We can get you close enough for your
purpose, no matter what that purpose may be.

One nice thing about the Fibonacci series is that it's known to converge rapidly,
so we can guarantee you that, say, the 20^th number divided by the 21^st number,
is pretty close to the ultimate limit of the series, and darn close to the answer to
your question.

That answer is 0.618033998521803 . (rounded)

If that's not close enough for you, we have yet another way to make you happy ...
a do-it-yourself kit that you can use to find the absolute ultimate limit of the
ratio of consecutive terms of the Fibonacci series. With this, you can find it just
as accurately as you need it, just by taking a simple square root. Here it is:

One term divided by the next one is

¹/₂ [ sqrt(5) - 1 ]

and one term divided by the last one is

 

¹/₂ [ sqrt(5) + 1 ].

These are exact, with no messy round-offs. Each is as accurate as the
square-root in it is.

You're quite welcome.