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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6answers
156 views

How to calculate $\sin(37°)$ with a Taylor approximation?

How to calculate $$\sin(37°)$$ with a Taylor approximation accurate to 3 decimal digits? I know it is not a difficult question, but I have no answers of my book and so far I have only determined the ...
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0answers
43 views

Can the Bernoulli numbers be viewed as a 'renormalization' of a finite geometric series with term $e^{-x}$, by integrating over $(-1,0)$?

I was playing around with ways to calculate Bernoulli numbers; for this post I will take their generating function as $x/(1-e^{-x})$, that is, $$ \sum_{n=0}^{\infty} \frac{B_n}{n!}x^n = \frac{x}{1-e^{-...
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1answer
24 views

What is a branch in Taylor polynomial (for someone with knowledge only in first-year calculus)?

I know that there has already many definitions of a branch, in the context of Taylor polynomial. Can someone describe to me, in a simplier term for someone with knowledge only in first-year calculus, ...
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1answer
36 views

Extrapolation using Taylor series - giving negative results for increasing positive inputs

I would like to use a 2nd order Taylor series expansion to perform an extrapolation to predict points outside of a known range. I am using the following formulation: \begin{equation*} d(N+ \Delta N) =...
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1answer
16 views

How to expand a local search from linear to higher order trajectories if higher order differentials are missing?

Consider $$f(x,y,k) = kx-y$$ and its level set $f(x,y,k) = 0$ Now for example assume that I have found a point on this level set $(9,81,9)$, how can I then estimate $k$ in $(8,76,k)$? This translates ...
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0answers
37 views

How many extra digits of precision do we need for $e^1$ so that $(taylor(e^1))^n = taylor(e^n)$

I'm trying to implement an algorithm to calculate $e^x$ for a given precision $p$ (that is $error < \frac{1}{10^p}$), using taylor series. $e^1$ converges pretty quick, so I'm wondering if for ...
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4answers
47 views

Using Taylor series to evaluate $\lim_{x\to0} \frac{\sqrt{1+ x\sin(x)} - \sqrt{\cos(2x)}}{ \tan^2(x/2)}$ [duplicate]

To be honest, I have to solve the following exercise, but I don't know what trigonometric formulas should I use (if I should), to get to a form where I can use the Taylor series. $$\lim_{x\rightarrow ...
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3answers
48 views

The process of using taylor series to evaluate limits.

For instance we want evaluate this simple limit using taylor series : $$L=\lim_{x\to 0}\frac{\sin x}{x^5}=\lim_{x\to 0}\frac{x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}+\cdots}{x^5}$$...
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2answers
61 views

Proof using taylor series

I am currently trying to solve a quantum mechanics problem in which i need to prove that $e^A e^{-A} = 1$ where $A$ is an operator and the exponent function is defined by a taylor series. However, I ...
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0answers
4 views

Truncation error of backward parabolic equation

I am trying to prove the backward parabolic equation truncate error but I don't know if it right hand the residual is clearly shown below. The function is : $$u_{t}(t_{n},x_{i})-u_{xx}(t_{n},x_{i}) = ...
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0answers
22 views

Is there a generalized solution to truncated exponential?

There's no closed form for the solution to all polynomials, but for certain classes it may be generalizable. For a table of polynomials $$\begin{align*} 1 \\ 1+t \\ 1 + t + \frac{1}{2}t^2 \\ 1 + t + \...
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2answers
73 views

Integral $x^{-x}\,dx$ using Taylor's?

So I'm good w my comparison tests but is this an elliptical function? $$ \int_0^int_12 x^{-x} \, dx$$ Does one use Taylor's integration in this?
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Verifying equalities of complex functions and series

I have an exercise that I don't know how to solve, I don't even know how to start to solve it: Let ${z \over exp(z)-1} = 1+B_1z+{B_2 \over 2!}z^2+···+{B_n \over n!}z^n+···$ a) Prove that the complex ...
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1answer
34 views

Quadratic approximation for a function to find a root [closed]

Let f: $\mathbb{R}$ → $\mathbb{R}$ be a twice differentiable function, to find a root x* of f, let consider a simple iteration on the quadratic Taylor polynomial; $q(x) = f(xk) + f′(xk)(x − xk) + (f′′(...
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0answers
27 views

what is the order of this polynomial approximation? [closed]

I have the polynomial : $1+x+\frac12x^2$ that is an approximation of $e^x$ around 0 and I have to find the order of this. I am a bit confused to find what it is, is it $2$ or $3$ and why ? Because of ...
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3answers
62 views

How to prove that $\sum _{n=0}^{\infty }\:\frac{(x^n)'}{(n-1)!} = e^{x}(x-1)$

I am trying to prove that $$\sum _{n=0}^{\infty }\:\frac{\left(x^n\right)'}{\left(n-1\right)!} = e^{x}(x+1)\tag 1$$ This sum is very similar to the derivative of exponential $(e^x)' = \sum _{n=0}^{\...
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1answer
35 views

Maclaurin series expansion 4th order

I'm trying to grasp how Maclaurin series with two variables develop into the forth order. I would appreciate if you correct my guess below. $$f(x,y)=f(0,0)+f_{1}(0,0)x+f_{2}(0,0)y\\ +\frac{1}{2!}(f_{...
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0answers
42 views

Is there a useful asymptotic expansion of $(1 + z^{\sqrt{2} - 1} + z^{\sqrt{2}})^{-1}$ at $z = 0$?

I believe that an asymptotic expansion in terms of powers of $z$ can not exist because we could use the geometric series to find something of the form $$ \frac 1{1 + z^{\sqrt 2 - 1} +z^{\sqrt 2 }} = \...
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2answers
32 views

Taylor series and Maclaurin Series, the center of the function and able to use the Maclaurin series instead.

I have been talking to a few people about when it is ok to use the Maclaurin series instead of using the Taylor series. I.e. Let $f(x)=e^{x^2-2x+1}$. Write down the degree 3 Taylor polynomial for $f(x)...
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1answer
20 views

Can I know if this equation can be well predicted by taylor polynomials?

I have the following equations, where the only unknowns are $x$ and $y$ and others are constants: $y = d+\frac{(-be)\pm e\sqrt{b^2 - 4ac + 4xc}}{2c}+\frac{f(2b^2 \mp 2b\sqrt{b^2-4ac+4xc}-4ac+4xc)}{4c^...
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0answers
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Power series for $\sqrt{1 + x}$ [closed]

How can I decompose a function $\sqrt{1 + x}$ into a power series for any x value?
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1answer
31 views

Imaginary asymptotics for the digamma function

I often see asymptotics and precise expansion for the gamma $\Gamma$ or the digamma $\psi$ function $\psi$ when the argument goes to $+\infty$, in particular when it stays real (or in a given angle ...
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1answer
15 views

Numerical analysis - Error analysis for the second order Taylor method

I'm trying to prove the following result for the second order Taylor method: $f$ is continuous and satisfies a Lipschitz condition with constant L on $D=\{(t,y)\mid a\leq t \leq b, -\infty<y<\...
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1answer
37 views

How to compute tangent of a matrix? [closed]

How to compute the tangent of a square matrix? Is there a method to compute as there exists for computing sine or cosine of matrix?
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1answer
73 views

Help me about $a_{n}−35(n+5)a_{n+3}+259(n^2+13n+5504)a_{n+6}−225(n+7)(n+8)(n+9)a_{n+9} =0$ [closed]

When I researched about Airy function, I reached the following equation $-225v^{'''}(t)+259t^{2}v^{''}(t)+(-35t^4+518t)v^{'}(t)+(t^{6}-70t^{3}+130)v(t)=0$ it is better that I solve this ordinary ...
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1answer
23 views

Hermite polynomial generating function

How would I write the following polynomial in terms of the Hermite polynomials, $H_n(z)$? \begin{equation} P_n(z) = \sum_{k=0}^{[n/2]} \frac{n!a^k}{k!(n-2k)!}(2a z)^{(n-2k)} \end{equation} I have ...
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1answer
42 views

How do I find the volume xyz tends to 0 when x(or y or z) tends to go $ \infty $

The volume V=xyz, and the constraint of surface area : S(x,y,z) = xy + yz + zx - 5 = 0, How do I comfirm the limit of volume V tends to 0 when x tends to go to $ \infty $? $$ \bbox[yellow] {\lim_{x\...
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1answer
11 views

How is the accuracy with which one function fits another numerically measured?

The accuracy with which a line fits discrete data points can be reflected with the Pearson correlation coefficient. But it is not always discrete data to which we try to fit a model function—for ...
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0answers
30 views

approximation of $\int \ln(1+\frac{2k}{a+b-k}) $

I'm trying to calculate an integration over an arbitrary polygon $P$, and a point $p$ is on $P$. $k$ is a constant, and $A(x,y,z)$ and $B(x,y,z)$ are arbitrary functions returning a scalar. As $\int_P ...
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4answers
64 views

Claculate limit $\lim_{x\to 0}\frac{1-(\cos(1-\sqrt{\frac{\sin(x)}{x}}))}{x^4}$

I have a problem to calculte this limit: $$\lim_{x\to 0}\frac{1-(\cos(1-\sqrt{\frac{\sin(x)}{x}}))}{x^4}$$ I used Taylor expansion for $\sin(x), \cos(x)$ and considered also $1-\cos(\alpha)=2\sin^2(\...
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0answers
13 views

Modelling functions using simultaneous polynomial equations.

Let $f(x) = \tan x$ under the interval $-π/2<x<π/2$. Let $x_k$ be the $k$-th value of $x$ under this interval. Consider an nth degree polynomial $$a_0x^n+a_1x^{n-1}... a_{n-1}x+a_n=0$$ Given ...
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1answer
61 views

Why can we not expand $(a+b)^n$ directly when $n$ is a fractional or negative index?

We know the binomial expansion of $(1+x)^n$ when $n$ is a fractional index or negative index. Why can we not expand $(a+b)^n$ directly when $n$ is a fractional or negative index? Instead of expanding ...
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1answer
14 views

the derivation of the Taylor expansion of the Frobenius norm $\|M-x x^{\top}\|_{F}^{2}$ with linear operator

Let $d$ be an integer. Let $[d]$ be $\{1,\dots, d\}$. For $\Omega\subset [d]\times [d]$, let $P_\Omega$ be the linear operator that maps a matrix $A$ to $P_\Omega (A)$, where $P_\Omega (A)$ has the ...
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1answer
39 views

Estimating the exp $exp(\bar x) - exp(\mu)$ with Taylor series

This is the Question! Please help :( There are 10 data, and their mean sd are $$ \bar x = 3.435, sd = 1.14965 $$ Jake wanted to estimate exp(mu) by the following estimator $$exp(\mu) ~ \approx exp(\...
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0answers
23 views

Perturb $\tanh(c(x-x_0))$ or similar functions around $x\rightarrow x+\delta x$

Given the perturbation $x\rightarrow x+\delta x$, how can the function $$ \tanh[c(x-x_0)], $$ $c$ and $x_0$ constant, be perturbed? I'd like to have a form like $$ f(x)\rightarrow f(x)+g(\delta x) $$ ...
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1answer
34 views

Taylor series expansion and value of cosine angle

This is a related question in deriving cosine values by 2 different means Deriving values of Trigonometric angles will be easier with Taylor series expansion for first few terms for some of the ...
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2answers
66 views

Exponential operator expansion

In my lectures, the professor discussed that for exponential linear operators it is $$ \exp(\lambda A + \lambda B) \neq \exp(\lambda A)\exp(\lambda B) $$ for $AB\neq BA$. Now I know that the ...
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0answers
11 views

Error of sum of taylor expansions

I am interested in the sum over composite functions, and would like to approximate the sum by the sum of the Taylor expansions (to first order to make it simple). Can I make a statement about the ...
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0answers
21 views

Find values such that $H_{0,1}$ is a Pade approximation of $H$

Consider a lathe with the dynamics: $mq''(t) + cq'(t) + kq(t) = −K(q(t) − q(t − τ )) + F$ Here, $q(t)$ denotes the position of the tool, whereas $m > 0$, $c > 0$, and $k > 0$ denote its mass, ...
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0answers
41 views

Euler-style sum of reciprocals of squares of roots of a function

If I understand correctly, historically, Euler proved that $$ \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} $$ by considering on the one hand taylor expansion of $$ \frac{\sin(x)}{x}=1-\frac{x^2}{...
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0answers
35 views

General form of the coefficients of $1/(f(1/z))$

Let $f(z) = z + \sum_{n=2}^{+\infty} a_n z^n$ be a holomorphic function on the unit disk. I'd like to compute the coefficients of $1/(f(1/z))$ which is a Laurent series. Of course, using the algorithm ...
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2answers
25 views

Maclaurin series decomposition limit

I need hint how to solve this limit using Maclaurin series. I'm confused, because couldn't find proper decomposition for denominator. $$ \lim _{x \rightarrow 1} \frac{e^{\frac{x-1}{x}}-\sqrt[4]{4 x-3}}...
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0answers
69 views

Finding the sum $\sum_{n = 1}^\infty \arctan\frac{2}{n^2}$ using Taylor series

I've been working with the series $$ \sum_{n = 1}^\infty \arctan\frac{2}{n^2}. $$ There is a nice solution to this using telescoping sums (Showing that $ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^...
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1answer
45 views

Series Expansion of $f(z) =$ $\frac{e^z}{z^3}$

I've come across this problem on my lecture book of Complex Variable Function, Expand $f(z) =$ $\frac{e^z}{z^3}$ into a power series on the $0 < |z| < \infty$ domain so far I've been learning ...
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1answer
22 views

Determining partial derivatives via Taylor polynomials

Typical Taylor series problem which has totally perplexed me. The function is $$e^{x_1x_2}\sin(x_1+x_2)$$ and the task is to solve $\partial_1^2\partial_2f(0,0)$ and $\partial_1\partial_2^3f(0,0).$ ...
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1answer
17 views

Help with Taylor Polynomial Estimation Solution.

Estimate $e^{0.1}$ to 6 decimal places using a Taylor polynomial about 0. Use error bounding to prove that your estimate is accurate to at least 6 decimal places. Is my solution correct? We know that ...
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0answers
14 views

Taylor expansion of exponential around 0 for $x = O(1)$

Let $x$ be some real value that is $O(1)$, find the taylor expansion of $e^tx$ at $0$. The text I am following writes: $$e^tx = 1 + tx + O(t^2x^2e^{O(t)})$$ I'm confused why there is an additional ...
3
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1answer
55 views

MacLaurin Series of $\tan(x)$

I am trying to compute the MacLaurin series of $\tan(x)$. I know this one is $$\tan x=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}$$ And I know how to derive this formula. ...
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0answers
7 views

Precision on two different approximations

Theoretically, how should I compare the precisions of the following two approximations? $f(x_2) = f(x_1) + (x_2-x_1)\times df(x_1)$ $f(x_2) = f(x_1) + (x_2-x_1)\times\frac{[df(x_1)+df(x_2)]}{2}$ where ...
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0answers
47 views

Behavior near isolated local minimum

Suppose $f: \mathbb{R}^n \to \mathbb{R}$ is smooth at its isolated local minimum $0$ and $f(0)=0$. Further suppose that there is some $N$ such that in any direction $u$, the $k$-th order directional ...

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