Question

If $1+2x+3x^2+4x^3+\dots \dots \infty \ge 4$, where $|x|<1$, then

• A. least value of $x$ is $\frac{1}{2}$
• B. greatest value of $x$ is $\frac{3}{2}$
• C. least value of $x$ is $\frac{2}{3}$
• D. greatest value of $x$ does not exists

Answer 1

Answer: (A), (D)

The correct options are
A least value of $x$ is $\frac{1}{2}$
D greatest value of $x$ does not exists

$S=1+2x+3x^2+4x^3+\dots \infty$
$\Rightarrow xS=x+2x^2+3x^3+4x^4+\dots \infty$
$\Rightarrow (1-x)S=1+x+x^2+\dots \infty =\frac{1}{1-x}$
$\Rightarrow S=\frac{1}{(1-x{)}^2}$

Now, $S\ge 4$
$\Rightarrow \frac{1}{(1-x{)}^2}\ge 4$
$\Rightarrow (x-1{)}^2\le \frac{1}{4}$
$\Rightarrow -\frac{1}{2}\le x-1\le \frac{1}{2}$
$\Rightarrow \frac{1}{2}\le x\le \frac{3}{2}$
$\Rightarrow \frac{1}{2}\le x<1(\because (0<|x|<1)$