#1736 · Algebra · Sequences & series

Infinite Product Coefficient

In the expansion of $(1 + x) (1 + a x) (1 + a^{2} x) (1 + a^{3} x) \dots$ , the number of factors being infinite, and $a < 1$ , shew that the coefficient of $x^{r}$ is $\frac{1}{( 1 - a ) ( 1 - a ^{2} ) ( 1 - a ^{3} ) \dots ( 1 - a ^{r} )} a^{\frac{1}{2} r (r - 1)}$ .

~15 minProof · checked by AIInfinite productsGenerating functions

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXII. Undetermined Coefficients, Examples XXII.b, No. 10

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