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Suppose I define y = Sum[Subscript[a, k] x^k, {k, 0, Infinity}] which is a power series representation for y. Now if I multiply y by bx^n+cx^m I want the resulting answer to be sum of two power series: Sum[b Subscript[a, k] x^(k+n), {k, 0, Infinity}] + Sum[c Subscript[a, k] x^(k+m), {k, 0, Infinity}]. However simply writing (bx^n+cx^m)y does not achieve this. How can I solve this problem?

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You can work with ReplaceAll rules

y = Sum[a[k] x^k, {k, 0, Infinity}]

y1 = y /. y[[1]] -> (y[[1]]*(b x^n + c x^m) // Expand)

enter image description here

y2 = y1 /. Sum[r_ + s_, t_] :> Sum@Sequence[r, t] + Sum@Sequence[s, t]

enter image description here

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