# 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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### Power series as approximation

I have to estimate the error when I approximate the function $$e^{\sin x}$$ to $$1+x+x^{2}+x^{3}$$ when $|x|<0.1$. I really don't know how to do because my teacher didn't teach me. But what I did ...
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### Find the Maclaurin series of f(x)=(arctan(x)-x)/x^3

What I think I need to to do is find a general series expansion of the function and then derive term by term to get the Macaurin series...but I'm not quite sure how to expand this function. Any help ...
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### Maclaurin polynomial of order 3? Order vs. Degree

I am doing some homework and came across a problem that asks: Find the Maclaurin polynomial of order 3 for f(x) = e^(-4x) When did some searching online, all searches came up as "...maclaurin ...
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### Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
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### Finding the Taylor series of arcsin(1-x)

I'm trying to calculate the Taylor series of $arcsin(1-x)$ about $x=0$. I'm having trouble because I can't compute the derivative there. I can see the correct solution on WolframAlpha (http://www....
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### Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
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### Confusion about the different ways of writing Taylor Polynomials

For the sake of using a simple example, let's say I want to approximate $y=x^3$ with a second degree polynomial, and let's say I want to construct my polynomial around the point $x=4$. One way I ...
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### Problem on series expansion and Bessel functions

One way to define Bessel functions is $$e^{\frac{x}{2}(t-\frac{1}{t})}=\sum_{n=-\infty}^{+\infty}J_{n}(x)t^n.$$ How do I prove that? I can't see a way of writing the L.H.S. as a geometrical (or ...
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### Finding the limit of: $\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$ using taylor polynomials

no solution provided so I was hoping someone would do a quick look over and make sure it looks ok. Finding the limit of: $$\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$$ ...
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### Find Polynomial of order 10 for $f(x)=sin(x)$ near x=0

My work so far : I presume the answer should look more like a summation? Thanks!
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### Compute $\frac{d^ny}{dx^n}$ if $y = \frac{7}{1-x}$

I am wondering what is $\frac{d^n}{dx^n}$ if $y = \frac{7}{1-x}$ Basically, I understand that this asks for a formula to calculate any derivative of f(x) (correct me if I'm wrong). Is that related to ...
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### Linearization of a function at a point

I have this delay differential equation $$\frac{dx}{dt}=a(x(t)-x(t-1))-b |x(t)|x(t)$$ and I have to make a linearization at the point $\left(\bar{x}(t),\bar{x}(t-1)\right)$, but I cannot figure out ...
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### Singular expansion of an implicit function

In the book of Flajolet and Sedgewick (this context is not so important, though), the following argumentation is used: Let $y(z)$ be a function given implicitly by $y - \phi(z,y) = 0$, where $\phi$ ...
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### How to solve asymptotic expansion: $\sqrt{1-2x+x^2+o(x^3)}$

Determinate the best asymptotic expansion for $x \to 0$ for: $$\sqrt{1-2x+x^2+o(x^3)}$$ How should I procede? In other exercise I never had the $o(x^3)$ in the equation but was the maximum order to ...
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### Implicit Euler using Taylor

I was reading script about differencial equatations. More specific about schemes that help calculate them - implicit Euler. That method was analyzed using something similar to Taylor but i am not sure ...
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### How to find similar convergence rates?

Consider the Taylor's series infinite summation of $\sin(x)$. Let $A_k=\sum\limits_{i=0}^k(-1)^i{x^{2i+1}\over (2i+1)!}$ (Series expansion of $\sin(x)$) I need a series $\{C\}_n$such that its ...
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### How to prove this Taylor expansion of $\frac{1}{(1+x)^2}=-1\times\displaystyle\sum_{n=1}^{\infty}(-1)^nnx^{n-1}$?

I came across this series of the Taylor Expansion- $$\frac{1}{(1+x)^2}=1 - 2x + 3x^2 -4x^3 + \dots.=-1\times\sum_{n=1}^{\infty}(-1)^nnx^{n-1}$$ But I have no idea how to prove this... Thanks ...
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### How do I apply a Taylor expansion of this?

Given $$\frac{1}{r}\left(1+\frac{2\epsilon \cos\theta}{r}\right)^{-1/2}$$ I was told by using Taylor expansion I could get $$1-\frac{2\epsilon \cos\theta}{r}$$ with term of order $\epsilon^2$. Can ...
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### How to derive a Taylor series from the ones we know ($\cos x$, $\sin x$, …)

If we know the Taylor expansion for the $\cos(x)$ function around $0$, how can we use it to derive the Taylor expansion of a similar function ($\cos(x+π/4)$) around $0$? I do know how to get the ...
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Let $v,w$ be vector fields and let $\Phi_t$ be the flow generated by $v$, that is $\frac{d}{dt}\big|_0\Phi_t(x)=v(x)$. The Lie derivative of $w$ in direction of $v$ is usually defined as $\mathcal ... 1answer 316 views ### Convergence of the quadratic map$\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$? Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the ... 1answer 39 views ###$\Pi_{n=0}^\infty (1-a_n)>0$if and only if$\sum a_n < \infty$. Let$a_n$be sequence in (0,1).$\Pi_{n=0}^\infty (1-a_n)>0$if and only if$\sum_{n=0}^\infty a_n < \infty$. First I considered$\sum log(1-a_n)$and tried to find sum inequality. I ... 2answers 49 views ### Finding certain coefficients in Taylor expansion of$ \log(1 +qx^2 + rx^3)$This exercise is part of the STEP$3$paper from$2014$. At a certain point in the problem, we 're supposed find$a_n$for$n = {2,5,7,9}$where$a_n$is the coefficient of$x^n$in the series ... 1answer 53 views ### Calculus of rank three tensor Let$A(\alpha)$be a matrix that depends to vector parameter$\alpha$. I want to approximate$A(\alpha+\Delta\alpha)$using Taylor expansion. My work: $$A(\alpha+\Delta\alpha) \approx A(\alpha)+\... 0answers 30 views ### Integration in an inequality Does integrating on both the sides of inequality with the same upper and lower limits with respect to same variable somehow affect the inequality. I saw an example lets say, Sin x < x ,x>0 ... 2answers 33 views ### How do I find the radius of convergence for \sum_{n=0}^{\infty}\frac{1}{\sqrt{n}}z^n? I'm a little unsure about methods on finding the radius of convergence of a function. It would be great to get some help on how to approach these kinds of problems. 1answer 40 views ### How to find Taylor series when x_0=0 and radius of convergence for \frac{x}{1+x} for f:(-1,\infty) Through the taylor series formula:$$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\dots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$I've got that f(x)=x-x^2+x^3-x^3\dots however my teacher claimed ... 1answer 31 views ### Does there exist a kernel concept for Taylor expansions? In the sense of kernel which one can use when weighing coefficients together in Discrete Fourier Transform (Dirichlet and Fejér kernels maybe most well known). Here are some examples of such kernels. ... 3answers 67 views ### Is the following is true? If that so, give me a proof. -log(1-x)=log(1+e^x)?? Is the following is true? If that so, give me a proof.$$-log(1-x)=log(1+e^x)?$$Give me some value where this equality holds. I dont think so it will be same. Because,$$(1-x)^{-1}=1+x+x^2+x^3+\... 3answers 183 views ### Elementary Proof of Ramanujan Master Theorem I was searching for an elementary proof of the Ramanujan Master Theorem and I found a page from Ramanujan's Notebook on wikipedia which contained the proof. I think that it has some gaps, so can ... 1answer 43 views ### How to find taylor polynomial of a function with two variables? Find the second order Taylor expansion about the point (1,-2) of the function$f(x,y) = (x^2 + y)e^{xy}$. I begin by computing the matrix of partial derivatives of f.$Df(x,y)=(2xe^{xy}+e^{xy}y(x^2+...
I have a series of dependent and independent variables. I would like to model their relation using a Volterra/Wiener series. How do I specify: $h_n$ $a$, $b$ Input vector $x_n$ The kernels for each ...