What's the completed table??
2 answers:
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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
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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Based on the equation given, the graph that matches the equation is Graph D.
<h3>How to find the graph?</h3>To find the graph that matches the equation, solve for the values of y based on values of x and the equation.
Assuming x is -1, y is:
= 1/5 (-1) + 2
= 1.8
x = 0
= 1/5(0) + 2
= 2
x = 1
= 1/5(1) + 2
= 2.2
The graph that has the points:
(-1, 1.8) (0, 2) and (1, 2.2) is the last graph so this is the correct graph.
Find out more on graphs at brainly.com/question/28747195
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