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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0
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1answer
24 views

Why does this work to shift a power series?

Problem: Find the Taylor series and the interval on which it is valid for $f(x) = \frac{1}{1-x}$ centered around $x=5$. The textbook's solution says to write $$\frac{1}{1-x} = \frac{1}{-4-(x-5)} = -\...
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0answers
19 views

Find Taylor series polynomial that gives uniform bound on error

The problem comes in two parts: Find an $\epsilon > 0$ such that for every $x\in[0,1]$ $$\left\lvert \sqrt{x}-\sqrt{x+\epsilon}\right\rvert \le \frac{1}{200}$$ We can show that $\left\lvert \...
0
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0answers
10 views

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 ...
0
votes
1answer
31 views

An alternative formula for a second order Taylor expansion?

I read in a book that the second order Taylor expansion of a function (around $x^0$) can be written as: $$f(x)=f(x^0)+\sum_{j=1}^n df(x^0)/dx_j*(x_j-x_j^0)+\sum_{j=1}^n\sum_{i=1}^nd^2f(x^1)/dx_idx_j*(...
0
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1answer
26 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 \...
0
votes
3answers
127 views

Why can't we do substitution in differentiation but is it ok in Taylor series?

I had the same question 10 years ago when I was studying high school. I didn't understand it and I gave up the math. 10 year ago, I needed to work with calculus during work and this question came to ...
0
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1answer
21 views

Tanh expansion problem

$$\frac{exp(-\beta e_k) - exp(\beta e_k)}{exp(-\beta e_k) + exp(\beta e_k)} = tanh(-\beta e_k)$$ In the context of mean field annealing I reached this equation but I am not sure how to expand tanh to ...
-4
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1answer
93 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.
2
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1answer
80 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)....
-1
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0answers
33 views

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 ...
1
vote
3answers
72 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 ...
2
votes
2answers
51 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,...
3
votes
1answer
66 views

What is the 2nd order taylor polynomial of f(x,y)?

I'm just computing the 2nd order taylor polynomial for $f(x,y) = tan(x + 3y + \frac{\pi}{4})$ centered at (3,-1) and wondering if I have done this correctly or if anyone has any suggestions on how I ...
0
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1answer
55 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, ...
0
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2answers
36 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}{...
1
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3answers
44 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}...
0
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1answer
21 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 $&...
0
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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 ...
1
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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 ...
1
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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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0answers
15 views

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 ...
1
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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....
2
votes
1answer
60 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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0answers
15 views

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 ...
0
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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 ...
2
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2answers
37 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 ...
2
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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: $$\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$$ ...
2
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1answer
33 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!
1
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2answers
514 views

Lagrange remainder vs. Alternating Series Estimation Theorem: do they always give you the same error bound?

Given a function and its nth degree Taylor series approximation, we can use the Lagrange form of the remainder to get a maximum value of the error of approximation. If the series is also an ...
4
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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 ...
0
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0answers
12 views

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 ...
1
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0answers
36 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 $\phi$ ...
2
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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 ...
0
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0answers
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 ...
0
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0answers
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 ...
3
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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 ...
1
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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-...
2
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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}{...
1
vote
1answer
28 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 ...
0
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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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2answers
36 views

Taylor Series in Fractional Calculus

I recently studied fractional calculus, namely the possibility to define fractional derivatives of some functions, like $$\frac{\text{d}^{1/2}}{\text{d}x^{1/2}}\ f(x) ~~~~~~~~~~~~~ \frac{\text{d}^{2/...
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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 }{(...
0
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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 ...
7
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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: $$\...
0
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1answer
27 views

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 ...
13
votes
1answer
313 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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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)+\...
1
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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 ...