#1770 · Algebra · Sequences & series

Scales of recurring series product

The scales of the recurring series $a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots$ and $b_{0} + b_{1} x + b_{2} x^{2} + b_{3} x^{3} + \dots$ are $1 + p x + q x^{2}$ and $1 + r x + s x^{2}$ respectively; shew that the series whose general term is $(a_{n} + b_{n}) x^{n}$ is a recurring series whose scale is $1 + (p + r) x + (q + s + p r) x^{2} + (q r + p s) x^{3} + q s x^{4}$ .

~20 minProof · checked by AIrecurring seriesscale of relationgenerating functions

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXIV. Recurring Series, Examples XXIV, No. 14

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