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 + …+ T_n  ; or,

o   T_m + T_(m+1) +…+ T_n

·        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 1^st ‘nomial’ (or, term) x  is in descending order of power from n à 0;

d.      The 2^nd ‘nomial’ y is in ascending order of power from 0 à n; the power of the 2^nd nomial plus 1 gives the position of the term in the series. Thus, in (x + y)ⁿ, term containing 2^nd nomial y  to the power 2 i.e. y² is the 3^rd 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, x^ay^b 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 = ⁿC_a = ⁿC_b

= 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 3^rd term (b = 2) = 15 = 6!/(4!2!)

·          The BC in 4^th term (b = 3) = 20 = 6!/(3!3!)

·          The BC in 5^th term (b = 4) = 15 = 6!/(2!4!)

·          The BC in 6^th term (b = 5) = 6 = 6!/(1!5!)

·          The BC in 7^th term (b = 6) = 1 = 6!/(0!6!)

·          In general, the BC of x^ry^(n-r) term in (x + y)ⁿ = n!/(r!(n-r)!) = ⁿC_r =

·          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) + ⁿC_nx⁰ 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 1^st nomial x = 1:

o  (1 + y)ⁿ: 1 + ⁿC₁y¹ + ⁿC₂y² +… + ⁿC_(n-1)y^(n-1) + ⁿC_nyⁿ; 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.      1^st nomial = 1,

                                                             ii.      modulus of 2^nd 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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