# Tagged Questions

Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

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### Taylor Expansion of $e^{itx}$, Expectation

the Taylor-expansion of $e^{itx}$ is $$1+itx+(itx)^2 / 2! + \cdots.$$ My question: Why can one write $1+itx+o(t)$ for the sum I sated above? $o(t)$ would mean that $(itx)^2 / 2! + \cdots$ would ...
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### Closed form for $n$-th derivative of $\sqrt{f(x)}$ for general $f(x)$

Let's assume we have an inifinitely differentiable real valued function $f(x)$, and we have a closed form expression for all its derivatives. Is it then possible to find a closed form for the $n$-th ...
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### Converge properties of Taylor Series expansion of complex function

I need to find the convergence properties of the Taylor Expansion of $$f(z)=\frac{z}{z-1}$$ I found the Taylor Series: $$\sum_{j=1}^\infty \frac{(-1)^{j+1}(z-i)^{j-1}}{(i-1)^j}$$ Then I used the ...
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### Inequality for a multivarialbe function?

For fixed $y\in \mathbb{C}^m$ and let $f$ be a fuction defined on $\mathbb{C}^m\times \mathbb{C}^m$ such taht $f(0,y)=1$ and $$\frac{\partial^n} {\partial x^n}f(x,y)=(i)^my^n{}f(x,y)$$ which means ...
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### How to expand the Taylor series of functions of several vectors?

We know that the Taylor series expansion of the function of several scalars around zero is $$f(x,y)=f(0,0)+f_x(0,0)\cdot x+f_y(0,0)\cdot y+\frac{1}{2!}f_{xx}(0,0)\cdot x^2+\dots$$ Then, how about ...