Sum $1 + 3 x + 6 x^{2} + 10 x^{3} + . . . .$ to infinity.

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Solution

Let $S = 1 + 3 x + 6 x^{2} + 10 x^{3} . . . . . . . \infty$
Multiplying both sides by $x,$ we get
$S x = 0 + x + 3 x^{2} + 6 x^{3} . . . . . . \infty$
Subtracting both eqns, we get
$S (1 - x) = 1 + 2 x + 3 x^{2} + 4 x^{3} . . . . . . . \infty$
Again multiplying by $x$ both sides, we get
$S x (1 - x) = 0 + x + 2 x^{2} + 3 x^{3} + 4 x^{4} . . . . . . . \infty$
Subtracting above eqns, we get
$S (1 - x)^{2} = 1 + x + x^{2} + x^{3} + x^{4} . . . . . . \infty$
Clearly above series is G.P. with common ratio $= x$
Then, $S (1 - x)^{2} = \frac{1}{1 - x}$
$\Rightarrow S = \frac{1}{(1 - x)^{3}}$

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1 + 3 x + 6 x^{2} + 10 x^{3} + . . . \infty

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