Egyptian-Fraction Expansions Of REALS

[Leroy Quet]

[posted also to sci.math]

An obvious idea I have not seen *much* about before (which means 
nothing...).


If we have a positive REAL, possibly irrational, x,

then we can find a sequence (many many sequences..., I bet) of positive 
integers
{n(k)}
such that

sum{k=1 to M} 1/n(k) = x,

where M is infinity if x is irrational.

For instance, after dealing with the integer-part differently, we can 
apply the Greedy-Algorithm to x.

x  =  pi,  as an example:

pi = 3 + 1/8 + 1/61 +...

Or x =  (sqrt(5)+1)/2 = phi:

phi  = 1 + 1/2 + 1/9 + 1/145 +...

(I am FAR from certain about the last terms in the above two expansions, 
since my calculator is low-precision.)
 

We might want to, in order to be strict with our definition of 
"Egyptian-Fraction", rewrite the "3"  in the pi-expansion as "1+1+1",  to 
get:

pi =  1 + 1 + 1 + 1/8 + 1/61(?) +...

In any case, I do not believe these sequences are in the Encyclopedia of 
Integer Sequences, although this idea seems basic to me.


I also wonder about alternative EF-expansions, both with all positive 
terms and with the possibility of having negative terms.

For instance, aside from:

pi^2/6  =  sum{k=1 to oo} 1/k^2,

what other expansions exist?

We could consider:

pi^2/6 = sum{k=1 to oo} (4/3)/(2k-1)^2,

but 4 does not divide (2k-1)^2, so we can rewrite this as:

pi^2/6   =
1/3 + 1/3 + 1/3 + 1/3 +
1/27 + 1/27 + 1/27 + 1/27 +
1/75 + 1/75 + 1/75 +1/75 +... 

So, therefore, if duplicate terms are allowed, then it is easy to see 
that it is possible for an  x  to have multiple expansions.

Anything more to be said about EFs of reals??
(I am sure there *must* be more to be said!...)

thanks,
Leroy
   Quet