Question

(1 point) Suppose that f(x) and g(x) are given by the power series f(x)=6+7x+3x2+5x3+⋯ and g(x)=7+8x+3x2+4x3+⋯. By multiplying power series, find the first few terms of the series for the product h(x)=f(x)⋅g(x)=c0+c1x+c2x2+c3x3+⋯.

Answer 1

Answer:

f(x).g(x)   =  c0+c1x+c2x2+c3x3+⋯.

$f(x).g(x) = 42 +97x +95x^{2} + 104x^{3} +77x^{4} +27x^{5} + 20x^{6}+....$

c₀ = 42 , c₁ =97 , c₂ = 95, c₃ =104, ...…...

Step-by-step explanation:

Suppose that f(x) and g(x) are given by the power series

f(x)=6+7x+3x2+5x3+⋯ and

g(x)=7+8x+3x2+4x3+⋯

By multiplying power series ,

f(x).g(x) = (6+7x+3x2+5x3+⋯).(7+8x+3x2+4x3+⋯)

           = 6(7+8x+3x2+4x3+⋯)+7x(7+8x+3x2+4x3+⋯)+3x2(7+8x+3x2+4x3+⋯) + 5x3(7+8x+3x2+4x3+⋯)+......

$f(x).g(x) = 42 +97x +95x^{2} + 104x^{3} +77x^{4} +27x^{5} + 20x^{6}+....$

f(x).g(x)   =  c0+c1x+c2x2+c3x3+⋯.is the form

$f(x).g(x) = 42 +97x +95x^{2} + 104x^{3} +77x^{4} +27x^{5} + 20x^{6}+....$

c₀ = 42 , c₁ =97 , c₂ = 95, c₃ =104, ...……