Concepts to remember on Binomial Expansions

1. Definition of binomial expansion: Binomial expansion of (a+b)^n expressed in powers and products of ‘a’ and ‘b’ is

$$(a+b{)}^n={}^n{C}_0{a}^n+{}^n{C}_1{a}^{n-1}b+{}^n{C}_2{a}^{n-2}{b}^2+{}^n{C}_3{a}^{n-3}{b}^3+\dots .+{}^n{C}_{n-1}a{b}^{n-1}+{}^n{C}_n{b}^n$$

2. General term of a binomial expansion: The rth term in the expansion of (a+b)^n is given by:

$$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$$

3. Pascal’s triangle and its properties:

• Pascal’s triangle is a triangular array of numbers that can be constructed by starting with a 1 at the top and then adding the two numbers above each entry to obtain the entry below.

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• The numbers in Pascal’s triangle are the binomial coefficients, which are the coefficients in the binomial expansion of (a+b)^n.
• The rth row of Pascal’s triangle gives the binomial coefficients for the expansion of (a+b)^n.

4. Applications of binomial theorem in approximation and integration:

• The binomial theorem can be used to approximate a function by a polynomial.
• This is done by taking the first few terms of the binomial expansion of the function.
• The binomial theorem can also be used to integrate some functions.
• This is done by using the formula:

$$\int (a+bx{)}^{n}dx=\frac{1}{b}[\frac{(a+bx{)}^{n+1}}{n+1}+C]$$ where C is the constant of integration.

5. Binomial theorem for negative and fractional indices:

• The binomial theorem can be extended to negative and fractional indices using the following formula:

$$(a+b{)}^{-n}=\frac{1}{(a+b{)}^n}=\sum _{r=0}^{\text{infin}}{}^{-n}{C}_r{a}^{n-r}{b}^r$$

$$(a+b{)}^{m/n}=\sum _{r=0}^{\text{infin}}{}^{m/n}{C}_r{a}^{m/n-r}{b}^r$$

$${}^{m/n}{C}_r=\frac{m(m-1)(m-2)\dots .(m-r+1)}{r!(n^r)}$$

• This formula can be used to expand any binomial expression with a negative or fractional index.

6. Convergence of binomial series:

• The binomial series (a+b)^n converges if and only if |a+b|<1.
• If |a+b|>=1, then the binomial series diverges.