Answer :
The first few term of the given power series are -
c₀ = 42, c₁ = 97, c₂ = 95, c₃ = 104 .....
Explain the term power series?
- A power series can be used to express any trigonometry, logarithmic, exponentially, geometric series, and polynomial function. Sin x, for instance, equals x - x3/3! + x5/5! - x7/7!
For the given condition-
f(x) and g(x) are given by the power series f(x)=6+7x+3x²+5x³+⋯ and g(x)=7+8x+3x²+4x³+⋯.
It is known that; f(x) = 6+7x+3+5+⋯ and g(x) = 7+8x+3+4+⋯.
Such that, estimated using following steps in to get the multiply the power series.
f(x) x g(x) = (6+7x+3x²+5x³+⋯ ) x (7+8x+3x²+4x³+⋯.)
f(x) x g(x) = 6(7+8x+3x²+4x³) + 7x(7+8x+3x²+4x³) + 3x²(7+8x+3x²+4x³) + 5x³(7+8x+3x²+4x³)
f(x) x g(x) = 42 + 97x + 95x² + 104x³ + 77x⁴ + 27x⁵ + 20x⁶ + ... ...eq 1
Now, the equation: gives the generalized power series.
h(x)=f(x)⋅g(x)=c₀+c₁x+c₂x²+c₃x³+⋯. ...eq 2
Compare eq 1 and eq2
c₀ = 42
c₁ = 97
c₂ = 95
c₃ = 104 .....
Thus, the first few term of the given power series are obtained.
To know more about the power series, here
https://brainly.com/question/14300219
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The complete question is-
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+⋯.
