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 Series coefficients

I have performed a Taylor Series expansion of a 2-D function in variables (y1,y2) and got something like: $f(y_1,y_2) = 3y_2 + 0.5y_1^2 + ...$ My question is that I would like this to "match" to ...
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1answer
12 views

Algebraic series, rational fraction of two variables in the form of polynomial

I come across the following claim: Let $y\in\mathbb{C}[[x]]$ be an algebraic series, that is, there exist $n\in\mathbb{N}^*$ $A_i(x)\in\mathbb{C}[x]$ for $i=0,...,n$ and $A_n(x)\neq 0$ such that \...
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3answers
69 views

Find the power series representation of $e^{-x^2}$

I know that the Maclaurin expansion of $e^x$ is $$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$$ But i'm not sure how to find the Maclaurin series here I tried this $$ f'_{(0)}=-2xe^{-x^2}=0 $$ And that ...
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2answers
49 views

Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives

I am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old series,...
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35 views

Computation of a complicated limit

Good morning to everyone! I don't know how to compute this type of limit... I got stuck at $arctan$. The limit is the following: $$ \lim _{x\to \infty }\left(\frac{\arctan \left(1-\cos \left(\frac{1}{...
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1answer
79 views

How to prove this integral equality?

The question is prove that$$ \lim_{n \to \infty} n^2 \left( \int_a^bf(x) \, \mathrm{d}x - \frac{b-a}{n} \sum_{i=1}^{n} f(a+(2i-1) \frac{b-a}{2n} )\right)= \frac{ (b-a)^2 }{24}\left( f'(b)-f'(a)\right)....
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3answers
39 views

Limit of the given expression.

For $x>0$,$$\lim_{x\to 0}((\sin(x))^{\frac{1}{x}}+(\frac{1}{x})^{\sin(x)})$$ is?. So now I calculated limits individually. Let $\lim_{x\to 0} ((\sin(x))^{1/x})=y$ thus I took log to get $\frac{1}{x}...
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1answer
19 views

Using Taylor Polynomial to Show How An Expression Of Only Real Numbers Can Be Approximated

I am studying for my graduate level GQE and looking at problems from old exams. The following question (from an unknown original source) reads: Suppose a,b,c and d are positive real numbers with a $&...
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2answers
50 views

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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2answers
41 views

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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0answers
25 views

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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1answer
89 views

new equation for $\int_0^ t e^{-x2} dx$? [closed]

fact! $$\int_0^ x e^{-x^2} dx$$ $$=e^{-x^2}\sum_{n=0}\frac{(2^n)x^{2n+1}}{{(2n+1)!!}}$$ Well the equation was new to me, when I derived by shear integration, and that is a cold HARD fact.
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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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1answer
52 views

(Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
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1answer
45 views

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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Multivariable taylor series approximation

The function is of the form $$ F(X) = sum_{i=0}^n x_i*(c_i + ln(x_i/xt)) $$ where $ X = (x_1,x_2,x_3,...,x_n) $ $ xt = sum_{i=0}^n x_i $ $ c_i $ is a constant term for ith species I want to find ...
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2answers
33 views

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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2answers
34 views

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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1answer
59 views

Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: \begin{equation} 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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1answer
32 views

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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2answers
77 views

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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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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35 views

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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2answers
46 views

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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3answers
76 views

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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16 views

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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17 views

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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2answers
64 views

Show that the Maclaurin series of $f(x)=1/\sqrt{1-x}$ holds for $x\in[0,1/2]$ using the Lagrange remainder theorem

Show that the Maclaurin series of $f(x)=1/\sqrt{1-x}$ holds for $x\in[0,1/2]$ using the Lagrange remainder theorem I can see that $$f'(x)=\frac12 (1-x)^{-\frac32}\text{ and }f''(x)=\frac12\frac32(1-...
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6answers
94 views

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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1answer
19 views

Computing the taylor series of $f(x)=(x+2)^{1/2}$ around $x=2$ up to order 2 terms using binomial series

I can expand $(x+2)^{1/2}$ by taking $y=x+1$ and using the binomial series to find a power series representation of $f$: $$\sqrt{y+1}=\sum_{k=0}^{\infty}{1/2\choose k}y^k=1+\frac{y}{2}+\frac{\frac{1}{...
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1answer
27 views

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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1answer
73 views

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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1answer
70 views

developing to maclaurin Series $f(x)=\frac{2x+3}{x^2 -4x+5}$ on $x=2$

$$f(x)=\frac{2x+3}{x^2 -4x+5}$$ on $x=2$. My solution: $t=x-2 $ => $x=t+2$ , we get: $f(t)=\frac{2t+7}{t^2+1}$ on $t=0$. then: $(2t+7)\sum_{n=0}^{\infty } {(-t^2)^n} = (2t+7)\sum_{n=0}^{\infty }{(...
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2answers
45 views

Show complex equation of closed curve integral

I need to show this equation: $$\frac{1}{2ia} \cdot \oint _{\gamma } \frac{e^{iz}}{z-ia}dz = \frac{e^{-a}}{2ia} \cdot \oint _{\gamma } \frac{1}{z-ia}dz$$ I have an hint to using Taylor. I have no ...
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4answers
103 views

Proof of Leibniz $\pi$ formula

I found the following proof online for Leibniz's formula for $\pi$: $$\frac{1}{1-y}=1+y+y^2+y^3+\ldots$$ Substitute $y=-x^2$: $$\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots$$ Integrate both sides: $$\...
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1answer
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Is it true that $(\Phi_{-t})_*w\circ \Phi_t=\sum_{n=0}\frac{t^n}{n!}\mathcal L^n_vw$?

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 ...
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1answer
37 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 ...
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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 ...
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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 ...
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2answers
32 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.
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1answer
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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 ...
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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+\...
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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)+\...
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1answer
40 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+...
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13 views

How do I specify the input to a Volterra series including the kernels?

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 ...
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1answer
28 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. ...
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25 views

How to find the asymptotic expansion of $\int_{-\infty}^{y} e^{-x^2/2}/\sqrt{2\pi} dx$ where $x \in N(0,1)$?

I realize the function inside the integral is the pdf of a normally distributed random variable x, but am unsure how to use this to solve the problem. I am trying to relate it to the inverse of the ...
2
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1answer
40 views

Multidimensional taylor series $sin (x^3y^2) $

A homework of mine was to compute the Taylor series of $f(x,y)=\sin(x^3y^2)$ around $(0,0)$ to the 25th order. I assumed, as $\sin(z)=\sum\limits^{\infty}_{k=0}(-1)^k\frac{z^{2k+1}}{(2k+1)!}$, that I ...
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2answers
42 views

Why my calculations aren't right? (Maclaurin series)

Good evening to everyone! I tried to calculate $ \cos\left( x- \frac{x^3}{3} + o(x^4)\right) $ using the MacLaurin series but instead of getting the final result equal to $1 - \frac{x^2}{2}+\frac{3x^4}...
2
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0answers
36 views

How to define hypergeometric function ${}_1 F_1(-n+1;-n+1;z)$ for $n$ positive integer

Consider a truncated Taylor series of the exponential function to approximate $e$: $$ E(n) = \sum_{k=0}^{n-1} \frac{1}{n!} $$ I thought of computing this using the hypergeometric finite series $_1 F ...