Mathematics: Find a power series representation for the function. (give your power series representation centered at x = 0.) f(x) = ln(5 − x) f(x) = ln(5) − ∞ n = 1 determine the radius of convergence, r. r =

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Find a power series representation for the function.

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dondre54

15.03.2022

Mathematics

If $f(x)=\ln(5-x)$ , then

$f'(x)=-\dfrac1{5-x}$

$f''(x)=-\dfrac1{(5-x)^2}$

$f'''(x)=-\dfrac2{(5-x)^3}$

and so on, following the general pattern

$f^{(n)}(x)=-\dfrac{(n-1)!}{(5-x)^n}$

Then the power series for f(x) centered at x=0 is

$\displaystyle f(0)+\sum_{n=1}^\infty\frac{f^{(n)}(0)}{n!}x^n$

$\displaystyle \ln5+\sum_{n=1}^\infty\frac{(n-1)!}{n!5^n}x^n$

$\displaystyle \ln5+\sum_{n=1}^\infty\frac{x^n}{n5^n}$

The power series converges by the ratio test for

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{(n+1)5^{n+1}}}{\frac{x^n}{n5^n}}\right|=\frac{|x|}5\lim_{n\to\infty}\frac n{n+1}=\frac{|x|}5$

or |x| , so that the radius of convergence is 5.

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Find a power series representation for the function. (give your power series representation centered at x = 0.) f(x) = ln(5 − x) f(x) = ln(5) − ∞ n = 1 determine the radius of convergence, r. r =

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