Series expansion for real number Suppose c is a real number with 0 < c < 1.  I am interested in the
following series.
c = 1/n1 + 1/n2 + 1/n3 + 1/n4 + .....
Where n1,n2, are positive integers chose to me as small as possible.  For
example
  4/5 = 1/2 + 1/4 + 1/20

 e-2 = 1/2 + 1/5 + 1/55 + 1/9999 + ....

By the way this expansion for e is accurate to 9 decimal places.
I have proved that if c = n/m then the series terminates before or after
n terms.  Note
6/109 = 1/17 + plus five more terms.  

The last term has 44 digits in the denominator.
I know that the integers n1, n2 satisfy the inequality
n(k+1) >= nk^2-nk + 1.

However, if n(k+1) = nk^2-nk+1 for all k=1,2,.... then the sum is
1/(n1-1) so it the series does not terminate you must have
   n(k+1) >= nk^2 -nk + 2 

infinitely often.  If you want to calculate pi to 1 trillion digits all
you need is the first 40 terms.  (I am not seriously suggesting this as
a way to calculate pi because finding the terms would be enormously more
difficult than calculating pi.)
I should warn you that making these calculations is very addictive
and I have already got another mathematician hooked.  Anyway, I find this expansion very interesting and I was wondering if this expansion has been discussed.
One question.  Suppose n1=2 and n(k+1)=nk^2-nk+2.  The sum
sum = 1/2 + 1/4 + 1/14 + 1/184 + 1/33674 + .... = 0.82689....

Is this a known number?  Thanks for listening.

This general type of representation (sum of fractions with numerator 1 and no repeat denominators) is called Egyptian fractions. You'll find a lot of further facts in the cited page. Knowing what to search for will net you lots of further tidbits.
The name is because pharaonic Egyptians used this (awkward to operate) way to represent fractions, with some exceptions like 2/3.
