Find the power series representation for g centered at 0 by differentiating or integrating the power series for f(perhaps more than once).

ka1leE 2021-03-07 Answered

Find the power series representation for g centered at 0 by differentiating or integrating the power series for f(perhaps more than once). Give the interval of convergence for the resulting series.
g(x)=ln(12x) using f(x)=112x

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Expert Answer

lobeflepnoumni
Answered 2021-03-08 Author has 99 answers

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 11y, setting y=2x. Its radius of convergence is 0.5
Let
f(x)=112x=1+(2x)+(2x)2+...+(2x)n+...
the power series expansion (geometric series),
valid for |2x|<1, |x|<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(12x)
=2dx(12x)=2f(x)dx
=2[1+(2x)+(2x)2+...+(2x)n+...]dx
=2x(2x)22(2x)33(2x)44...(2x)nn...
is the power series expansion for g(x),
radius of convergence =0.5

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Jeffrey Jordon
Answered 2021-12-16 Author has 2064 answers

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