M2. Binomial Theorem, Exponential and Logarithmic Series:

1. Binomial Theorem:

For any positive integer $n$; $x$ and $a$ are any number. Then,

$(a + x)^n = C(n,0)a^n + C(n,1)a^{n-1}\;x + C(n,2)a^{n-2}\;x^2 + ...... + C(n,n)x^n$ .......... (i)

To proof - By actual Multiplication:

$(a + x)^2 = a^2 + 2ax + x^2$
                $= C(2,0)a^2 + C(2,0)a\,x + C(2,2)x^2$

$(a + x)^3 = a^3 + 3a^2x + 3ax^2 + x^3$
                $= C(3,0)a^3 + C(3,1)a^2\,x + C(3,2)a\,x^2 + C(3,3)x^3$

Similarly, let us assume that for the some particular value $n$ is

$(a + x)^n = C(n,0)a^n + C(n,1)a^{n-1}\;x + C(n,2)a^{n-2}\;x^2 + .......$
                      $+ C(n,r)a^{n-r}\;x^r + ........ + C(n,n)\;x^n$  ...................................... (ii)

This equation (ii) is the general expansion of Binomial expansion.

» General term of Binomial expansion:

The term $(r+1)^{th}$ in the binomial expansion of $(a + x)^n$ called general term, because any required term may be obtained from it, denoted by $t_{r+1}$ and defined as

$(a+x)^n = \sum_{r=0}^{n}C(n,r)a^{n-r}\;x^r$ ............................ (iii)

For the $(r+1)^{th}$ term;

$t_{r+1}$ = $C(n,r)a^{n-r}\;x^r$ .............................. (iv)

» Middle term of Binomial Expansion:

(i) When $n$ is even, we can write $n = 2r$. Then there is only one middle term $(r +1)^{th}$ is,

$t_{\frac{n}{2}+1}$ = $C(n,r)\;a^{n-r}\;x^r$ = $C(2r,r)\;a^r\;x^r$ .......................... (v)

(ii) When $n$ is odd, we can write $n = 2r - 1$. Then there will be two middle terms as follows:

$t_r$ = $C(2r-1,\;r-1)\;a^r\;x^{r-1}$ ...................... (vi)

and

$t_{r+1}$ = $C(2r-1,\;r)\;a^{r-1}\;x^{r}$ ....................... (vii)

» Some properties of expansion, when $n$ is any $+ve$ integer.

i) The number of terms in the expansion is $(n+1)$. 

ii) In each term, the sum of the exponent is $n$. 

iii) The expansion starts with the $1^{st}$ term $a^n$, and end with the last terms $x^n$.
(Where we find the exponent of $a$ decreases by one and that of $x$ increases by one). 

iv) The coefficient of the terms equidistant from the beginning and the end are always equal.

» Binomial Coefficient:

From the equation (i), the coefficient $C(n,0)$, $C(n,1)$, $C(n,2)$ .......... $C(n,n)$ in the expansion of $(a + n)^{x}$ are known as binomial coefficients. Which are respectively denoted as $C_0$, $C_1$, $C_2$ ............. $C_n$.

For $a$ = 1; in equation (ii) we have,

$(1 + x)^n = C_0 + C_1\;x + C_2\;x^2 + C_3\;x^3 +....... + C_{n-1}\;x^{n-1} + C_n\;x^n$ ............ (viii)

2. Exponential and Logarithmic Series:

i) An exponential function is a function of the form: 

$y = f(x) = b^{x}$;  $b>0$ .....................(i)

where base $(b)$ is a constant and the index $(x)$ is a variable.

As a function of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (i.e. its derivative) is directly proportional to the value of the function.

» Expansion of $e^x$:

$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + .......... +\; \frac{x^r}{r!} \;+ ........ to \;\infty$        ........................ (ii)

$e^{-x} = 1 - \frac{x}{1!} + \frac{x^2}{2!} - \frac{x^3}{3!} + .......... +\; (-1)^r\;\frac{x^r}{r!}\; + ........ to \;\infty$   ................ (iii)

ii) The inverse of an exponential function is called a logarithmic function. If $y = f(x) = b^{x}$ be a function; Then in the form of logarithmic: $x = log_{b}\;y$

» The expansion Logarithmic Series:

$log_{e}(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + .................$         .................. (iv)

$log_{e}(1-x) = - x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - ..............$             ................... (v)

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