The sum of the series 1 + 3x + 6 x^2 + 10 x^3 + ........ will be - Vidyadip

MCQ

The sum of the series $1 + 3x + 6{x^2} + 10{x^3} + ........\infty $ will be

A
$\frac{1}{{{{(1 - x)}^2}}}$
B
$\frac{1}{{1 - x}}$
C
$\frac{1}{{{{(1 + x)}^2}}}$
✓
$\frac{1}{{{{(1 - x)}^3}}}$

✓

Answer

Correct option: D.

$\frac{1}{{{{(1 - x)}^3}}}$

d
(d) Let $S = 1 + 3x + 6{x^2} + 10{x^3} + .....\infty $

$ \Rightarrow $$x.S = x + 3{x^2} + 6{x^3} + .......\infty $

Subtracting $S(1 - x) = 1 + 2x + 3{x^2} + 4{x^3} + .......\infty $

$ \Rightarrow $$x(1 - x)S = x + 2{x^2} + 3{x^3} + .......\infty $

Again subtracting,

$ \Rightarrow $$S[(1 - x) - x(1 - x)] = 1 + x + {x^2} + {x^3} + ........\infty $

$ \Rightarrow $$S[(1 - x)(1 - x)] = \frac{1}{{1 - x}} $

$\Rightarrow S = \frac{1}{{{{(1 - x)}^3}}}$

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