The coefficient of x^{2012} in frac{1+x}{(1+x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $f(x) = \frac{1+x}{(1+x^2)(1-x)}$.Using partial fractions,we write $\frac{1+x}{(1+x^2)(1-x)} = \frac{Ax+B}{1+x^2} + \frac{C}{1-x}$.$1+x = (Ax+

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(A) Let $f(x) = \frac{1+x}{(1+x^2)(1-x)}$.Using partial fractions,we write $\frac{1+x}{(1+x^2)(1-x)} = \frac{Ax+B}{1+x^2} + \frac{C}{1-x}$.$1+x = (Ax+B)(1-x) + C(1+x^2) = Ax - Ax^2 + B - Bx + C + Cx^2$.Comparing coefficients:$x^2: -A + C = 0 \implies A = C$.$x^1: A - B = 1$.$x^0: B + C = 1$.Substituting $A=C$ into $A-B=1$ gives $C-B=1$. Adding $B+C=1$ gives $2C=2$,so $C=1$ and $A=1$. Then $B=0$.Thus,$f(x) = \frac{x}{1+x^2} + \frac{1}{1-x} = x(1+x^2)^{-1} + (1-x)^{-1}$.Using the binomial expansion $(1+z)^{-1} = \sum_{n=0}^{\infty} (-1)^n z^n$ and $(1-z)^{-1} = \sum_{n=0}^{\infty} z^n$:$f(x) = x \sum_{n=0}^{\infty} (-1)^n x^{2n} + \sum_{n=0}^{\infty} x^n = \sum_{n=0}^{\infty} (-1)^n x^{2n+1} + \sum_{n=0}^{\infty} x^n$.We want the coefficient of $x^{2012}$.The first sum only contains odd powers of $x$,so the coefficient of $x^{2012}$ is $0$.The second sum is $\sum_{n=0}^{\infty} x^n$,so the coefficient of $x^{2012}$ is $1$.Total coefficient $= 0 + 1 = 1$.

The coefficient of $x^{2012}$ in $\frac{1+x}{(1+x^2)(1-x)}$ is

Similar Questions

If $\frac{2x^4-3x^2+4}{(x^2+1)(x^2+2)} = a + \frac{px+q}{x^2+1} + \frac{mx+n}{x^2+2}$,then $\frac{n}{q} =$

The partial fraction of $\frac{x^2}{(x - 1)^3(x - 2)}$ is

If $\frac{x^4}{(x-1)(x-2)(x-3)}=p(x)+\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x-3}$ then $p\left(\frac{3}{2}\right)+C=$

$\begin{aligned} & \text{If } \frac{4x^2+5x^4+7}{(x^2+1)(x^4+x^2+1)} = \frac{Ax+B}{x^2+1} \\ & + \frac{Cx^3+Dx^2+Ex+F}{x^4+x^2+1}, \text{ then } \\ & B+2(D+F+E)-C \cdot A = \end{aligned}$

Given $\frac{3x-2}{(x+1)^2(x+3)} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x+3}$,find the value of $4A + 2B + 4C$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module