### Binomial Expansions

**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

$$(a + b)^n =\ ^{n}C_0 a^n + \ ^{n}C_1 a^{n-1}b +\ ^{n}C_2 a^{n-2}b^2 + \ ^{n}C_3 a^{n-3}b^3 +….+\ ^{n}C_{n-1}ab^{n-1}+\ ^{n}C_{n}b^{n}$$

**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.

$$ \hspace{2.5cm}\boxed{\hspace{0.3}1\\hspace{0.3}1\hspace{0.3}1\\hspace{0.3}1\hspace{0.3}2\hspace{0.3}1\\hspace{0.3}1\hspace{0.3}3\hspace{0.3}3\hspace{0.3}1\\hspace{0.3}1\hspace{0.3}4\hspace{0.3}6\hspace{0.3}4\hspace{0.3}1}$$

- 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:

$$\int (a+bx)^n dx = \frac{1}{b}\left[ \frac{(a+bx)^{n+1}}{n+1}+C\right]$$ 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:

$$(a+b)^{-n} = \frac{1}{(a+b)^n} = \sum_{r=0}^\infin \ ^{ -n}C_r a^{n-r}b^r$$

$$(a+b)^{m/n} = \sum_{r=0}^\infin \ ^{m/n}C_r a^{m/n-r}b^r$$

$$ ^{m/n}C_r =\frac{m(m-1)(m-2)….(m-r+1)}{r!(n^r)}$$

- 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.