The power series is
g(x) = -2x - (2x)2/2 - (2x)3/3 -(2x)4/4 -.......-(2x)n/n - .....
To deduce the power series of g(x) from the power series for f(x) and identify its radius of convergence
The power series for f(x) is just the geometric series derived from 1/1-y ,setting y=2x.
Its radius of convergence is 0.5
Let,
f(x)= 1/1-2x = 1+ (2x) + (2x)2 + .........+(2x)n......+....
The power series expansion (geometric series),
valid for I2xI < 1 , IxI < 0.5
so, radius of convergence = 0.5
The power series for g(x) is found by integrating term by term the power series of f(x) (upto a constant). The radius of converngence of g(d) is the same as that of f(x) (from general theory) =0.5
Now, g(x) = ln(1-2x)
= -2 
=-2 ![\int\limits^a_b {[1+(2x)+ (2x)2 +........+ (2x)n+.......]} \, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%5Ea_b%20%7B%5B1%2B%282x%29%2B%20%282x%292%20%2B........%2B%20%282x%29n%2B.......%5D%7D%20%5C%2C%20dx)
g(x) = -2x - (2x)2/2 - (2x)3/3 -(2x)4/4 -.......-(2x)n/n - .....
is the power series expansion for g(x).
radius of convergence =0.5
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Answer:
Answer: Using the proportions between the segments, the area of the triangles are:
1. Area of triangle ADC is 25 in^2.
2. Area of triangle BDC is 15 in^2.
3. Area of triangle CDE is 12 in^2.
Step-by-step explanation:
You would be able to find this out using the calculator
9514 1404 393
Answer:
p(x) = x^4 -5x^3 +20x -16
Step-by-step explanation:
If 'a' is a zero of the polynomial, then (x -a) will be a factor. For the given zeros, the simplest polynomial will be the product of the corresponding factors:
p(x) = (x -2)(x +2)(x -4)(x -1) . . . . . . note that x -(-2) = x +2 (factored form)
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Multiplying these out gives the result in standard form.
The product of the <em>first two factors</em> is a "special product" recognizable as the difference of two squares.
(x -2)(x +2) = x^2 -2^2 = x^2 -4
The product of the <em>last two factors</em> can be found in the usual way. The distributive property applies.
(x -4)(x -1) = x(x -1) -4(x -1)
= x^2 -x -4x +4 = x^2 -5x +4
Then the full polynomial is the product of these partial products:
p(x) = (x^2 -4)(x^2 -5x +4)
= x^2(x^2 -5x +4) -4(x^2 -5x +4)
= x^4 -5x^3 +4x^2 -4x^2 +20x -16
p(x) = x^4 -5x^3 +20x -16 . . . . . . . . standard form
