# 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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### higher order asymptotic expansion for likelihood ratio

I have been studying Hayakawa(1975) and (1977) and was wondering if anyone has already computed higher order terms for his expansions following his framework. I'd be very happy if someone could ...
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### Show that $\tfrac{(-1)^{n}}{\cos(n)+n^{\tfrac{3}{4}}}=\tfrac{(-1)^{n}}{n^{\tfrac{3}{4}}}+\mathcal{O}\left( \tfrac{1}{n^{\tfrac{3}{2}}}\right)$

I would like to show that : $$\dfrac{(-1)^{n}}{\cos(n)+n^{\tfrac{3}{4}}}=\dfrac{(-1)^{n}}{n^{\tfrac{3}{4}}}+\mathcal{O}\left( \dfrac{1}{n^{\tfrac{3}{2}}}\right)$$ by starting from the left side ...
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### How would Taylor Series work?

I wish to calculate sine of any given an angle without using the functions that come with programming language and devices. I have written a small code in Python which can be found here. Using the ...
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### Taylor expansion $f(x)=f(0)$

The following taylor expansion of the function $f(x)$, requires $f(x)$ to have a derivative up to what order? $$f(x)=f(0)+f'(0)x+f''(0)x^2/2+\mathcal{O}(x^3)$$ My solution: Based on the Taylor'...
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### Show that $(-1)^{n}\sqrt[n]{n}\sin(\frac{1}{n})=\tfrac{(-1)^{n}}{n}+\mathcal{O}\left(\tfrac{\ln(n)}{n^{2}} \right)$

I would like to show that : $$(-1)^{n}\sqrt[n]{n}\sin(\frac{1}{n})=\dfrac{(-1)^{n}}{n}+\mathcal{O}\left(\dfrac{\ln(n)}{n^{2}} \right)$$ by starting from the left side and get the right side : My ...
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### Show that $(-1)^{n}\left(\sqrt{n+1}-\sqrt{n} \right)=\tfrac{(-1)^{n}}{2\sqrt{n}}+\mathcal{O}\left(\tfrac{1}{n^{\frac{3}{2}}} \right)$

I would like to show that : $$\fbox{(-1)^{n}\left(\sqrt{n+1}-\sqrt{n} \right)=\dfrac{(-1)^{n}}{2\sqrt{n}}+\mathcal{O}\left(\dfrac{1}{n^{\dfrac{3}{2}}} \right)}$$ by starting from the left side ...
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### Taylor polynomial in function composition

I have the Taylor polynomial of a function f(x): $$4-5x+2x^2$$ and the Taylor polynomial of a function g(x): $$2+\frac{1}{2}x-\frac{1}{8}x^2$$ Both about $$x=0$$ How can I calculate the Taylor ...
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### Taylor expansion of Crystal Field potentials

I am trying to work through Michael Tinkham's "Group Theory and Quantum Mechanics". In discussing crystal field theory he uses the following example: We start with an atom at the origin. We want to ...
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### Does the Taylor series of $e^{f(x)}$ converge everywhere?

In STAT 110, the professor says "the Taylor series of $e^x$ converges everywhere, and then proceeds to convert: $${e}^{t^2/2} = \sum_{i=0}^\infty \frac{{(t^2/2)}^n}{n!}$$ I understand that the ...
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### How to determine the truncation error with products and quotients

If I have an equation given by $$\displaystyle Y = \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$ and I expand $a,b,c,d$ in a Taylor series, where $a$ is truncated at the $A^{th}$ order, $b$ is ...