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mf1021
find the radius of convergence of the taylor series around x=0 for ln(1/(1+2x))
i spose we would nee to generate the power series for it first
\[[ln\frac{1}{1+2x}]'=\frac{1/(1+2x)'}{1/(1+2x)}\] \[[ln\frac{1}{1+2x}]'=\frac{-2/(1+2x)^2}{1/(1+2x)}\] \[[ln\frac{1}{1+2x}]'=\frac{-2/(1+2x)}{1/1}\to\ -\frac{2}{1+2x}\]
lol -ln(1+2x) = -2/(1+2x)
-2+4x-8x^2+16x^3-32x^4+64x^5 ... ----------------- 1+2x ) -2 (-2-4x) 4x (4x+8x^2) -8x^2 (-8x^2-16x^3)
somewhere i had seen something like this integration dln(1+2x)/dx and expand it as power series.
soo, a relevant power series would be:\[ln(\frac{1}{1+2x})=\int\sum (-1)^{n+1}\ 2^{2n}x^{n}dx\]
\[ln(\frac{1}{1+2x})=\sum_{0}^{inf}\frac{(-1)^{n+1}}{n+1}x^{n+1}\]
\[ln(\frac{1}{1+2x})=\sum_{0}^{inf}\frac{(-1)^{n+1}2^{n+1}}{n+1}x^{n+1}\] \[lim\frac{(-1)^{n+1}2^{n+1}x^{n+1}}{n+1}\frac{n}{(-1)^{n}2^{n}x^{n}}\] \[lim\frac{-2x}{n+1}\frac{n}{1}\] \[|x|lim\frac{-2n}{n+1} = -2\]
prolly shoulda taken the - with the x :)
all non - ns should be vacated \[|-2x|\ lim\frac{n}{n+1}=2x\] \[2x<1;\ x<\frac{1}{2}\]