If $| x | < 1$ then the co-efficient of $x^{n}$ in the expansion of $(1 + x + x^{2} + . . . . .)^{2}$ .

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Solution

We've for $| x | < 1$ , $(1 - x)^{- 1} = 1 + x + x^{2} + . . . . . . . .$ .......(1).
Now,
$(1 + x + x^{2} + . . . . . . .)^{2}$
$= (1 - x)^{- 2}$ [ Using (1)]
$= 1 + 2 x + 3 x^{2} + . . . . . + (n + 1) x^{n} + . . . . . .$ .
So, co-efficient of $x^{n}$ is $(n + 1)$ .

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