## Thursday, 14 July 2016

### Binomial Series / Binomial Theorem

My Personal Notes on Binomial Series / Binomial Theorem:

1.      Meaning of Binomial Series:
·        A binomial is a mathematical expression which has 2 different terms, such as in: (a + b)0; (x + y)1; (x + 2y)2; (1 + x)-1; (1 – 2x)-1;…………. (1 + 3x2)(-1/4); (1 + x)5; (1 + x/√3)10; (1 + 3x)(1/5); (1 – 3x/4)(1/3);…
·        Terms in series are given as:
o   T1 + T2 + T3 + …+ Tn  ; or,
o   Tm + T(m+1) +…+ Tn
·        Binomial series refers to the series of terms created from the expansion of a binomial to its power (or, exponent or index).

2.      Binomial Expansion Leads to Binomial Series with Binomial Coefficient for Each Term in the Series:
Expanding these binomial leads to these series of term(s) and coefficients:
Terms                                          Bi. Coefficients
·        (x + y)0:       1x0y0 = 1                                                   1
·        (x + y)1:                 x + y                                            1     1
·        (x + y)2:            x2 + 2xy + y2                                  1    2     1
·        (x + y)3:        x3 + 3ax2y + 3axy2 + y3                    1   3     3    1
·        (x + y)4:    x4 + 4x3y + 6x2y2 + 4xy3 +y4            1    4    6    4    1
·        (x + y)5: …                                                      1   5     10   10   5   1
·        (x + y)6: …                                                    1   6  15   20   15  6   1
·        (x + y)n: xn + (n/1!)x(n-1)y1 + ((n)(n-1)/2!)x(n-2)y2 + …+ yn

As can be seen from the above:
a.       Binomial expansion to power n produces (n + 1) number of terms in the binomial series;
b.      For each and every term in the series, the sum of powers of x and y is n
c.       The 1st ‘nomial’ (or, term) x  is in descending order of power from n à 0;
d.      The 2nd ‘nomial’ y is in ascending order of power from 0 à n; the power of the 2nd nomial plus 1 gives the position of the term in the series. Thus, in (x + y)n, term containing 2nd nomial y  to the power 2 i.e. y2 is the 3rd term of the binomial series (3 = 2 +1).
e.       The binomial coefficients form what is called ‘Pascal Triangle’ (Pascal: 1623 – 1662) or ‘Yang Triangle’(Yang Hui: 1261):
i.      Each line of the Δ starts and ends with 1
ii.      Each line is symmetrical about its middle
iii.      Each line of binomial coefficients (BC) is obtainable from the binomial coefficients in the line above it – by adding the top-flanking BC (binomial coefficients).
iv.      Hence, when the power of expansion n is not too large, say, n < 8, Pascal or Yang Δ can be used to quickly find out the binomial coefficients of the terms

3.      Usefulness and Limitation of Pascal or Yang Triangle
a.       Useful to get binomial coefficients when power n is small (say, n < 8)
b.      Difficult to do so when power n is high

4.      Limitation of Pascal Triangle Overcome by Binomial Theorem
a.       Pascal discovered (in fact, Yang Hui discovered about 450 years earlier than Pascal) what is known as Binomial Theorem as follows: They discovered that the binomial coefficient (BC) of any term in a binomial series, say, xayb term in (x + y)n can be expressed as:

BC = n!/(a!b!)

Where,
·        a + b = n
·        n! is read as n-factorial
·        n! = n(n -1)(n – 2)…(3)(2)(1)
1! = 1
2! = 2 x 1
3! = 3 x 2 x 1
4! = 4 x 3 x 2 x 1
5! = 5 x 4 x 3 x 2 x 1
6! = 6 x 5 x 4 x 3 x 2 x 1
·        0! is defined as: 0! = 1

Since,               a + b = n,
therefore:          a = (n – b)       ;or,       a! = (n – b)!
or,                    b = (n – a)       ;or,       b! = (n – a)!

Hence,      BC = n!/(a!b!)        Or,    BC = nCa = nCb
= n!/[(n-b)!b!] =
= n!/[a!(n-a)!] =

b.      Discovery of Binomial Theorem:
Now, look at the BC in this series:

(x + y)6 = x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6

You will discover that:
·          The BC in 3rd term (b = 2) = 15 = 6!/(4!2!)
·          The BC in 4th term (b = 3) = 20 = 6!/(3!3!)
·          The BC in 5th term (b = 4) = 15 = 6!/(2!4!)
·          The BC in 6th term (b = 5) = 6 = 6!/(1!5!)
·          The BC in 7th term (b = 6) = 1 = 6!/(0!6!)
·          In general, the BC of xry(n-r) term in (x + y)n = n!/(r!(n-r)!) = nCr =

·          Thus, generally, binomial expansion can be expressed as:
o  (x + y)n: nC0xny0 + nC1x(n-1)y1 + nC2x(n-2)y2 +… + nC(n-1)xy(n-1) + nCnx0 yn; or
o  (x + y)n: xn + x(n-1)y1 + x(n-2)y2 +… + xy(n-1) + yn;
o  (x + y)n: xn + n!/((n-1)!1!)x(n-1)y1 + n!/(n-2)!2!)x(n-2)y2 + …+ yn
o  (x + y)n: xn + (n/1!)x(n-1)y1 + ((n)(n-1)/2!)x(n-2)y2 + …+ yn

·          Special case where 1st nomial x = 1:
o  (1 + y)n: 1 + nC1y1 + nC2y2 +… + nC(n-1)y(n-1) + nCnyn; or
o  (1 + y)n: 1 + y1 + y2 +… + y(n-1) + yn;
o  (1 + y)n: 1 + n!/((n-1)!1!)y1 + n!/(n-2)!2!)y2 + …+ yn
o  (1 + y)n: 1 + (n/1!)y1 + ((n)(n-1)/2!)y2 + n(n-1)(n-2)/3!y3…+ yn

c.       Conditions for binomial series (1 + x)n  to be convergent:
i.      1st nomial = 1,
ii.      modulus of 2nd nomial, lxl < 1,
iii.      n is rational
hence, these binomial series are convergent:
·        (1 + x)n : IF lxl < 1, in other words:  -1 < x < 1
·        (1 – 2x)-1: IF l-2xl < 1 or lxl < ½ i.e -1/2 < x < ½
·        (1 + 3x2)(-1/2): IF l3x2l < 1; lxl < 1/3(1/2);  -1/3(1/2) < x < 1/3(1/2)

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