Questions tagged [taylor-expansion]

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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Proving two polynomials converge to the same function

I am looking to create polynomials that converge past the usual radius of convergence on the real line. So far, I have proved that this polynomial will converge to the analytic continuation of the ...
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Taylor Series representation of $f(x) = \sqrt{x} + \frac{1}{\sqrt x}$ at $a=1$

I am trying to find the Taylor Series representation of $f(x)= \sqrt x + \frac1{\sqrt x}$ at $a = 1.$ With $5$ terms. I know how to get the series expansion. centered at $a=1$. with $5$ terms… However ...
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What are first order residual, second order residual, third order residual…

What are first-order residual, second-order residual, third-order residual...and so on. Like the difference between the original function (evaluated at some value) and the approximated function (to ...
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What is $\lim \limits_{n\to\infty }\frac{n^x}{n!},\ n\in \mathbb Z^+,\ x\in \mathbb R$? Does $n^x$ or $n!$ grow faster as $n\to \infty$?

The question just pops up in my head. $n^x$ is just an exponential function with positive integer base. My intuition is that $n^x$ grows faster so the limit should be infinity. How do we properly ...
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Is taking a Taylor expansion of $x^{x^x}$ possible?

Disclaimer: I do not know that much math, so there might be obvious mistakes that I make in this question. As far as I can tell, taking the derivative of $x^{x^x}$ infinitely is possible. However, ...
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show that x-x^(3)/6 < sinx< x for x>0. ( taylor series)

the question in a random study problem says show that x - $\frac{x^3}{6} < sinx< x$ for $x>0$. ( taylor series). I am assuming the question means about x=0 I have tried finding the maximum ...
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Finding the norm of the Taylor approximation of a multivariate function $f:\mathbb{R}^n\mapsto\mathbb{R}^m$

I have a function $f:\mathbb{R}^n\mapsto\mathbb{R}^m$. My goal is to bound the first order Taylor approximation of $f$. Given $x,x'\in\mathbb{R}^n$ I have that f(x)-f(x')\approx (x-x')...
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Not sure where Term comes from in Expansion of Geometric Sequence

Quick question - I'm just wondering how that $\mu_1$ term appeared in the second equality next to the $x_1$ term.
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Functions as Taylor coefficients

I'm stuck at the following lemma. Let $f:[a,b]\to\mathbb{R}$, and suppose that there exists $\phi_1,...,\phi_n:[a,b]\to\mathbb{R}$ continuous functiosn such that: f(x+h) = f(x) + \...
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How does one treat big O when Taylor expanding $\sin(\sin x)$?

I am working on finding the Taylor (Maclaurin expansion) of $\sin(\sin(x))$ to the third order. We have $$\sin{t} = t - \frac{t^3}{6} + \mathcal{O}(t^5)$$ If I then set $t = \sin(x)$, and expand once ...
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Show that $x-x^3/6<\sin x<x$ for $x>0$ [closed]

the question in a random study problem says show that $x-x^3/6<\sin x<x$ for $x>0$. (taylor series). I am assuming the question means about $x=0$ I have tried finding the maximum error ...
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Find the Taylor series of $(1-x^2)/(4-x)$ [closed]

i got this question on my test and so far I have no idea how I should use approach that problem. hope some of you could help me
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Why does the Maclaurin serie of $(1+x)^k$ converge only for $|x|< 1$?

So I understand that if $f(x)=(1+x)^k$ you get something like \begin{align} f(x)=(1+x)^k \Rightarrow f(0)=1\\ f'(x)=k(1+x)^k \Rightarrow f'(0)=k\\ \vdots\\ f^{(n)}(x)=k(k-1)\dots (k-n+1)x^{k-n} \...
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Determine the limit using Taylor expansions:

I struggle with this one—maybe someone could point me in the right direction. $$\lim_{x\to 0} \frac{5^{(1+\tan^2x)} -5}{1-\cos^2x}$$ Getting the Taylor series expansion for $\tan^2x$ and $\sin^2x$ is ...
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Is it useful to expand the likelihood in Taylor serie *without using* the log likelihood ? Does it make sense?

The log likelihood has the advantage of transforming the product in sum. The likelihood is often expanded in a Taylor serie around the best solution, by using the log likelihood in the Taylor ...
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Taylor expansion of $\frac {1}{|x-y|}$with x and y two vectors

This equation comes from a physics script on electrodynamics, saying that this equation comes from a Taylor series expansion. I understand the first equality, but not the second one. It is really not ...
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Converse of Taylor expansion theorem

The Taylor theorem states, that if a function $f$ is $n$-times differentiable at point $a$, it has a unique expansion $$f(x) = \sum_{k=0}^n f^{(k)}(a) \frac{(x-a)^k}{k!} + o((x-a)^n)$$ Suppose now we ...
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Is the $\xi_n$ of the Lagrange's remainder of $\log(1+x)$ monotonical?

This question is a special case of my previous one, Are there any $f(x)$ whose $\xi_n$ of the Lagrange's remainder does not converge to $0$?. Consider the Maclaurin's series of $\log(1+x)$ with ...
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Taylor Expansion for $\ln\sqrt{x^2-2x+2}$ at $x_0=1$. [closed]

I have just started to learn Taylor expansions. I am confused in this question. Can you explain its Taylor Expansion at $x_0=1$ clearly? $\ln\sqrt{x^2-2x+2}$ at $x_0=1$. Thanks.
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Taylor expansion for $\frac{x-1}{x^2-2x+2}$ at $x_0=1$.

I have just started to learn Taylor expansions. I am confused in this question. Can you explain its Taylor Expansion at $x_0 = 1$ clearly? $$f(x) = \frac{x-1}{x^2-2x+2}.$$
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Approximating using Taylor series expansion

Using linearisation (truncated Taylor series expansion) around 0, show that: $1-2 \omega \sin ^{2}\left(\frac{k \pi h}{2}\right)$ $\approx$ $1-\frac{\omega k^{2} \pi^{2}}{2} h^{2}$ I am a bit lost on ...
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Finding the best Taylor-esque polynomial

Background Based only on the information provided by derivatives at a single point, the Taylor series provides the 'best'* approximation of the function around its radius of convergence. But, some ...
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