Mathematics: Find a power series representation for the function and determine the interval of convergence ln(5 - x

Find a power series representation for the function

Ответ:

datgamer13

17.01.2022

To find the power series representation of In(5 - x), we recall that the power series representation of a function of the form:
$\frac{a}{1-r} =\sum\limits^\infty_{n=0}ar^n$
provided |r| < 1
Also recall that
$\int{\frac{a}{1-r}\,dx} =\int{\sum\limits^\infty_{n=0}ar^n\,dx}$
Notice that
$\ln{(5-x)}=-\int {\frac{1}{5-x}\,dx}$
To get the power series of
$\frac{1}{5-x}= (\frac{1}{5}) \frac{1}{1- \frac{x}{5} } \\ =\sum\limits^\infty_{n=0}( \frac{1}{5}) (\frac{x}{5})^n=(\frac{1}{5}) (\frac{x}{5})+(\frac{1}{5}) (\frac{x^2}{25})+(\frac{1}{5}) (\frac{x^3}{125})+ . . . \\ =\frac{x}{25}+\frac{x^2}{125}+ \frac{x^3}{625}+ . . .$
Therefore, the power series representation of
$\ln{(5-x)}=-\int {\frac{1}{5-x}\,dx} \\ =-\int{(\frac{x}{25}+\frac{x^2}{125}+ \frac{x^3}{625}+ . . . ),dx} \\ =C-\frac{x^2}{50}-\frac{x^3}{375}- \frac{x^4}{2500}- . . .$
When x = 0: C = ln 5
Therefore,
$\ln{(5-x)}=\ln{5}-\frac{x^2}{50}-\frac{x^3}{375}- \frac{x^4}{2500}- . . .$

The radius of convergence is given by |r| < 1.
Here,
$r= \frac{x}{5}$
Therefore, radius of convergence is
$| \frac{x}{5}| \ \textless \ 1 \\ |x|\ \textless \ 5$

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Ответ:

oliviaicono14

06.03.2020

Mathematics

The modern Russians formed from two groups of East Slavic tribes: Northern and Southern.

Explanation:

The area that is today the country of Russia has been inhabited by people for thousands of years. The first modern state in Russia was founded in 862 by King Rurik of the Rus, who was made the ruler of Novgorod. Some years later, the Rus conquered the city of Kiev and started the kingdom of the Kievan Rus.

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Find a power series representation for the function and determine the interval of convergence ln(5 - x

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