[Math] Taylor expansion at $x=0$ of $\ln(1/(1-x))$

calculustaylor expansion

Hello I am having some trouble with the taylor expansion of
$$f(x)= \ln \frac1{1-x}$$

Would it be correct to treat the inner part as the following geometric series? $$\sum_{k=0}^\infty x^k$$

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First I would write $$f(x) = \ln\frac{1}{1-x} = - \ln(1-x)$$ now differentiate to find: $$f'(x) = \frac{1}{1-x}$$ Now use the geometric series to expand the derivative.

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[Math] Taylor Expansion of Error Function

Elaborating a little on Marty's comment gives the following:

$f^{(n)}(a)$ can be written in terms of Hermite polynomials $H_n$:

$$ H_0(x)=1,\, H_1(x)=2x,\, H_2(x)=4x^2-2,\, H_3(x)=8x^3-12x,\, H_4(x)=16x^4-48x^2+12,\, H_5(x)=32x^5-160x^3+120x,\, H_6(x)=64x^6-480x^4+720x^2-120,\dots\, $$ You recognize that $H_{2n-1}(0)=0$, which gives the power series for $e^{-x^2}$ at $a=0$: $$ e^{-x^2} = 1 - \frac{2}{2!}x^2+\frac{12}{4!}x^4-\frac{120}{6!}x^6+\cdots $$ (see here). After multiplying by $2/\sqrt{\pi}$, this integrates to $$ \operatorname{erf}(z) =\frac{2}{\sqrt{\pi}} \left(z-\frac{z^3}{3}+\frac{z^5}{10}-\frac{z^7}{42}+\frac{z^9}{216}-\ \cdots\right) . $$

EDIT: Since $\displaystyle \frac{d^n}{dx^n}e^{-x^2}= (-1)^n e^{-x^2} H_n(x), $ one can do a Taylor Series for every $a$: $$ \text{erf}_a(x)= \text{erf}(a) + e^{-a^2} \sum_{n=1}^\infty (-1)^n \frac{H_n(a)}{n!}(x-a)^n $$

[Math] Can a function be approximated by finite number of Taylor expansion terms outside of disk of convergence

No. Once you restricted yourself to a finite number of terms, you have a polynomial. A polynomial $p(z)=\sum_{n=0}^d c_n z^n$ can be expanded in terms of $(z-a)$ for any point $a$ (just by using the binomial formula), but this does not change the polynomial: you still have the same values $p(z)$ at every $z$, just written differently. So, if the polynomial $p$ did not do a good job of approximating $f$ around $a$, the algebraic manipulations won't change that.

The magic of analytic continuation is possible because reshuffling pieces of an infinite series can change its behavior.

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[Math] Taylor Expansion of Error Function

[Math] Can a function be approximated by finite number of Taylor expansion terms outside of disk of convergence

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