# Power series multiplication

Please show proper notation, justification and step by step work. n See attachment for problem

Given that the zeros for (sinx)/x are the values x=0, x= +-pie, x=+-2pie, x=+-3pie,.... (x=0 must be excluded, why?)

This implies that F(x) can be factored as follows

F(x) = (1-(x/pi)) (1-(x/-pi)) (1-(x/2pi)) (1-(x/3pi)) (1-(x/-3pi))....

= [(1-(x/pi))(1+(x/pi))] [(1-(x/2pi))(1+(x/2pi))] [(1-(x/3pi))(1+(x/3pi))....

= (1-((x^2)/(pi^2))) (1-((x^2)/(4pi^2))) (1-((x^2)/(9pi^2))) (1-((x^2)/(16pi^2))) (1- ((x^2)/(25pi^2)))...

Multiply out this infinite product (or at least the first 6 factors) and collect like terms in x's to form a polynomial.

i.e F(x)= 1- (???????)x^2 + (????????)x^4 -(???????)x^6 +........

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Please see the attachment.

Let . Since , so is not a zero of . To make , we must have ...

#### Solution Summary

This is a problem regarding power series multiplication.