Respuesta :
Answer:
f(x).g(x) = c0+c1x+c2x2+c3x3+⋯.
[tex]f(x).g(x) = 42 +97x +95x^{2} + 104x^{3} +77x^{4} +27x^{5} + 20x^{6}+....[/tex]
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+⋯)+......
[tex]f(x).g(x) = 42 +97x +95x^{2} + 104x^{3} +77x^{4} +27x^{5} + 20x^{6}+....[/tex]
f(x).g(x) = c0+c1x+c2x2+c3x3+⋯.is the form
[tex]f(x).g(x) = 42 +97x +95x^{2} + 104x^{3} +77x^{4} +27x^{5} + 20x^{6}+....[/tex]
c₀ = 42 , c₁ =97 , c₂ = 95, c₃ =104, ...……
The value of [tex]\rm c_0 = 42[/tex], [tex]\rm c_1 = 97[/tex], [tex]\rm c_2 = 95[/tex], [tex]\rm c_3 = 104[/tex] and so on and this can be determined by using the arithmetic operations.
Given :
f(x) and g(x) are given by the power series f(x)=6+7x+3x2+5x3+⋯ and g(x)=7+8x+3x2+4x3+⋯.
Given that f(x) = 6+7x+3[tex]\rm x^2[/tex]+5[tex]\rm x^3[/tex]+⋯ and g(x) = 7+8x+3[tex]\rm x^2[/tex]+4[tex]\rm x^3[/tex]+⋯. So, the following calculation can be used in order to multiply the power series.
[tex]\rm f(x)\times g(x) = (6+7x+3x^2+5x^2+..)\times (7+8x+3x^2+4x^3+..)[/tex]
[tex]\rm f(x)\times g(x)=6(7+8x+3x^2+4x^3)+7x(7+8x+3x^2+4x^3)+3x^2(7+8x+3x^2+4x^3)+5x3(7+8x+3x^2+4x^3)+...[/tex]
[tex]\rm f(x) \times g(x) = 42+97x+95x^2+104x^3+77x^4+27x^5+20x^6+...[/tex] --- (1)
Now, the generalized power series is given by the equation:
[tex]\rm h(x) = f(x) \times g(x) = c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5+c_6x^6+...[/tex] ---- (2)
Now comparing equation (1) and equation (2).
[tex]\rm c_0 = 42[/tex]
[tex]\rm c_1 = 97[/tex]
[tex]\rm c_2 = 95[/tex]
[tex]\rm c_3 = 104[/tex] and so on.
For more information, refer to the link given below:
https://brainly.com/question/13101306
