Question

The sum of the infinite series $1+3x+4x^2+10x^3+18x^4+....$ at $x=\frac{1}{3}$ is

• A. 28/9
• B. 25/8
• C. 49/16
• D. none of these

Answer 1

The correct option is D none of these

$S=1+3x+4x^2+10x^3+18x^4+....$ ---------- (1)

$2x\times S=2x+6x^2+8x^3+20x^4+....$ ----------- (2) --- multiplying by 2x

Subtracting (2) from (1), we get

$=(1-2x)S=1+x-2x^2(1-x+x^2-x^3.....)$

$=(1-2x)S=\frac{(1+x{)}^2-2x^2}{1+x}$ [since 0$<$x$<$1 and sum of infinite series]

Now substituting $x=\frac{1}{3}$ we get $S=\frac{7}{2}$