The theorem is called binomial because it is concerned with a sum of two numbers (bi
means two) raised to a power. Where the sum involves more than two
numbers, the theorem is called the Multi-nomial Theorem. The Binomial
Theorem was first discovered by Sir Isaac Newton.
We can write a Binomial Coefficient as: [0.1]
Similarly, combinations can be written: [0.2]
"n" as a nonnegative integer
The Binomial Theorem or Formula, when n is a nonnegative integer and k=0, 1, 2...n is the kth term, is: [1.1]
k>n, and both are nonnegative integers, then the Binomial Coefficient
is zero. This explains why the above series appears to terminate. That
is, it has (n+1) terms.
It can also be written: [1.2] Where n and k are references to numbers in Pascal's Triangle, is called a Binomial Coefficient and is read as "n over k". When it is a combination, it may be read as "n choose k".
Proof of the Binomial Theorem
Binomial Theorem was stated without proof by Sir Isaac Newton
(1642-1727). The Swiss Mathematician, Jacques Bernoulli (Jakob
Bernoulli) (1654-1705), proved it for nonnegative integers. Leonhart
Euler (1707-1783) presented a faulty proof for negative and fractional
powers. Finally, the Norwegian mathematician, Niels Henrik Abel
(1802-1829), proved the theorem.
Proof when n and k are positive integers
When n and k are nonnegative integers, then the Binomial Coefficients in: [1.2, repeated]
be considered combinations, and read "n choose k", as appropriate. Sir
Isaac Newton just rote the formula down in his notebook, without proof,
perhaps because he thought the formula was self-evident.
Looking at 1.2, we choose the number of x's, and first we choose none at all. n choose 0 is 1.
Similarly, we choose 1, 2, ... n x's.
we choose 1 from n, this is the combination n choose 1, which we know
is n. When we consider the x-squared term, we choose 2 x's from n, and
the formula for this is n(n-1)/2. The combinations appear to prove the theorem.
Another approach, is to assume [2.1] where the subscripts are simply for labelling the factors.
The first term of this expansion will be an,
and the coefficient is 1. We will think about choosing x's. So the
first terms, which uses all the a's and none of the x's is the number
of ways we can choose 0 x's from n, which is only one way.
The next term is the x term.
We need to choose 1 x from n possibilities, and this is n ways. The
coefficient of x is therefore n. This will be of the power (n-1). The
sum of the powers of the n's and a's will always be n.
The x2 term,
is made up of choosing one x and then choosing another (so they make
x-squared). One way to think of this is to pick the first x from the
first factor, and then we have a choice of (n-1) factors for the next
x. We can choose the first x in n ways and there are (n-1) other
choices for the next x. There are therefore n(n-1)/2! ways of choosing
two x's and this is the x2 coefficient. The factorial 2
comes about in the following way. Consider factors 1 and 2. We can
choose an x from factor 1 and then one from factor 2, or we can choose
an x from factor 2 and one from factor 1. Clearly, these are
effectively the same so we are double counting. So we divide by
factorial 2. (This is the number of permutations of 2 objects).
x3 term is made up of three x's. We have a choice of n x's for the
first x, but once we have chosen, we have only (n-1) x's remaining for
the second x. And for the third x we have (n-2) choices. This means we
can choose three x's in n(n-1)(n-2)/3! ways and this is the coefficient
of x3. The factorial 3 makes allowance for identical
permutations. With three things, we can arrange them in 6 ways, but,
for our purposes, these three x's, irrespective of the order of
selection, are the same combinations. For instance, we could choose
three x's from the first three factors in 6 ways: 123, 132, 213, 231,
312, 321, but they all make up the same x3!
general, we can select k x's from n factors in n(n-1)(n-2)...
(n-k+1)/k! ways. And this is the coefficient of the general term.
have proved the Binomial Theorem for nonnegative integers n and k,
essentially by creating its terms and showing they are the same as the
terms claimed for the Binomial Theorem.
Proof when r is any real number
now prove the Binomial Theorem when the power r, is any real number:
positive, negative, rational, irrational, fractional.. any real number.
I have use (1+x) instead of (a+x) for simplicity, and because we
often use the Binomial Theorem in this way. If there is an a, we simply
take it out of the brackets. The following proof uses
simple calculus, and the proof rests on the truth of simple
calculus. It assumes that a Binomial Expansion can be written as: [4.1] where r is a real number and k is an integer. ak are the coefficients of the expansion.
When x=0, then a0=1
We differentiate and get: [4.2] When x=0, a1=r
Differentiating again and setting x=0 [4.3] a2=r(r-1)/2
is evident that we are extracting the Binomial Coefficients. To cut a
long story short, let us differentiate the function k times [4.4] Setting x=0 and rearranging: [4.5]
Taking the general term (4.5), we show that the left-hand side equals: [4.6]
the right-hand side is the Binomial Theorem! We have therefore proved
the Binomial Theorem for all real numbers, so we can legitimately use
it with positive and negative and fractional r, and we are no longer
limited to integers. The method of expanding (1+x)r is
known as a Maclaurin Expansion. The Binomial Theorem is so versatile
that x can even be a complex number, with a non-vanishing imaginary
Defining the Binomial Coefficients
When n and k are nonnegative integers, we can define the Binomial Coefficients as: [6.1] k
is positive and less than n. If k is less than zero, we have negative
factorials, which we haven't defined (see using the Gamma Function to define factorials,
for a method to define factorials for fractions and complex numbers.
Factorials of the negative integers do not exist.) When k is greater
[6.1] is zero, as expected. (This is what makes the Binomial Expansion
with n as a nonnegative integer terminate after n+1 terms!)
When r is a real number, not equal to zero, we can define this Binomial Coefficient as: [6.2] When r is zero, [6.2] gives zero instead of 1, so we restrict [6.2] to r≠0. We define B(n,0) as 1.
If the Binomial Coefficient is also a combination (n and r are positive integers), then we can use the rules of combinations.
Sum of Binomial Coefficients
We can write the binomial theorem as:
Where n is a positive integer, and k is a nonnegative integer, 0, 1, ..., n and is the term number. If we let a=b=1, we find (1+1)n=2n
is the sum of the terms, because the powers of a and b are all 1, and
only the coefficients remain. Naturally, the values of the coefficients
are not changed by the values of a and b, so the sum of the
coefficients is always 2n, whatever the values of a and b.
If we write a=1 and b=-1, then (1-1)n=0. We also notice that the even powers of b will be positive and the odd powers will be negative.
sum of the terms of a binomial expansion equals the sum of the even
terms (and the even powers of b), k=0, 2, etc plus the sum of the odd
terms, k=1, 3, 5, etc:
Because when a=1 and b=-1, the odd terms and the even terms cancel out, and their coefficients are therefore equal, we have:
coefficients do not change with the values of a and b, so the sum of
the coefficients of the odd terms is always equal to the sum of the
coefficients of the even terms, when n is a positive integer.
the sum of the coefficients of the even terms equals the sum of the
coefficients of the odd ones, and because the total sum is 2n, we have the sum of the even terms and the sum of the odd terms are both equal to half the total sum: ■
the Binomial Expansion is finite, when r is a nonnegative integer, then
the series is always convergent, being the finite sum of finite terms.
It is when the series is infinite that we need to question the when it
converges. According to the ratio test for series convergence a series converges when: [7.1] It diverges when: [7.2] And the result is indeterminate when: [7.3] For the Binomial Theorem the ratio of the kth and the k-1 terms is: [7.4]
the ratio test when the Binomial Expansion is an infinite series (r is
not a nonnegative integer), we find the limit is [7.5]
That is, |x|. The Binomial Theorem converges when |x|<1.
When |x| is 1, the ratio test does not advise us on its status. For example: [7.6]
is an infinite series. When x=1, the left-hand side is 1/2 and the
right-hand side is 1-1+1-1+... Yet it appears to be divergent, in the
sense of meaning not convergent (that is, it does not converge to a
single finite value), because it seems to oscillate between -1, 0 and
(It seems that the answer 1/2 may be correct, however) When x=-1, the
left-hand side is 1/0, which is infinite, and the right-hand side is
1+1+1... which is also clearly infinite. In this case, the series
clearly diverges. When |x|=1, we need to examine these cases very
carefully. The simple answer, however, is that when |x|=1, the binomial
series is indeterminate, so discussing the value when x=1 is
meaningless.1/(1+1) is a half, but we cannot obtain this from the Binomial Theorem.
The conclusion here is that when the binomial series is infinite (n is negative or fractional), then it converges when |x|<1