Binomial Theorem Derivation
Name: Ken M.
In every textbook of mathematics or work on the history
of mathematics, Sir Isaac Newton's expression for the binomial theorem in
negative powers is merely stated, not derived. How did Newton derive the
binomial theorem for negative exponents? I have not found the answer in
any reference work.
The binomial expansion works for positive, negative, fractional, and even
complex exponents although the formulas get a bit long. Two on-line
references will guide you through the process for the non-positive cases.
The first: http://hyperphysics.phy-astr.gsu.edu/hbase/alg3.html#bef gives
a short explanation so that you can convince yourself "it works". The
second: http://www.escape.com/~paulg53/math/pi/newton/ gives the
derivation in all its messiness. If you still need some more cases to fill
in between these two extremes, I suggest you do a "Google" search on the
search term: "binomial theorem negative exponents". You will find many
"hits" with all levels of generality and messiness. You are correct in
your comment that the derivation for other than positive exponents is left
as "One can show..." or "It is left as an exercise for the student..."
that mathematicians are famous for!
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Update: June 2012