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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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maclaurin series for sin(2x): show that it converges to sin2x for all x.

if the function were sinx we can prove that the error term tends to zero as the degree of the polynomial tends to infinity. however, with sin 2x the (n+1)th derivative is $$2^{n+1} (sin x )or (cos x)$...
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23 views

$u(x)=c \cdot \lambda (\frac{x}{2})$ for $c$

Given: Function $u(x)$ is infinitely differentiable and $\lambda \in \Bbb{R}$ equation ${\bigcirc}\hspace{-4mm}{1}\space$: $u(x)= \lambda \cdot u(\frac{x}{2})$ find all $\lambda$, for which the ${\...
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44 views

Taylor series for $e^{-(x-t)^2}$

I can't seem to see why the Taylor series for $e^{-(x-t)^2}$ is as follows $$ e^{-(x-t)^2} = \sum_{n=0}^{\infty} \frac{(-t)^n}{n!} \left( \frac{d}{dx} \right)^n e^{-x^2} $$
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How do I determine the interval over which my error calculation should be conducted?

I've been instructed to find the values of x for which the function $f(x) = e^{-2x}$ may be approximated by the Maclaurin series $1-2x+2x^2-\frac{4}{3}x^3$ with an error of less than 0.001, but no ...
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Determining the values for which a Maclaurin Polynomial approximates $f(x)=\cos(x)$ within an error of 0.001

I'm tasked with determining the values of $x$ for which $f(x)=\cos{x}\approx 1-\frac{x^2}{2!}+\frac{x^4}{4!}$ has an error no greater than 0.001. Using the error for Taylor polynomials $E = |R_n(x)| =...
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Taylor Series and hyperbolic area of a triangle

a) Express cos(A) in terms of cos(α) If area of a hyperbolic triangle is π - α - β - γ, then area of a hyperbolic equilateral triangle is π - 3α. In order to get cos(A) in terms of cos(α), cos(A) = ...
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25 views

Finding the first three nonzero terms in the Maclaurin series: $y=\frac{x}{\sin(x)}$

As the title says I would like to find the first three nonzero terms in the Maclaurin series $$y=\frac{x}{\sin(x)}$$ I have the first few terms for the expansion for $\sin(x)=x-\frac{x^3}{6}+\frac{x^...
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how to prove an f^2n(x) equation when given f(x) using maclaurin series? [on hold]

So I have found the equation for f^n(0) = (1/n!)*x^2n. Then I plug in 2n for n, but this is where i get confused. That would cause the equation to be 1/(2n)!x^4n and when you plug in 0 for x you get 1/...
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How do I write the Taylor expansion of this function $z = f(x,y)$

I have an assignment where I need to write the Taylor expansion for the function $$ z = f(x,y)$$ which is given by the following formula: $$ z^3 − 2xz + y = 0$$ for the point $A(1,1)$. I know how ...
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Approximating square root for long expressions

I'm currently working on a problem which asks me to calculate the potential energy of a three spring system arranged in an equilateral triangle constrained to move in the x-y plane. As a consequence ...
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1answer
45 views

Different forms of Maclaurin expansion of $f(x,y) = \frac{1}{x+y}$

I am trying to use Taylor series to approximate the value of $f(x,y) = \frac{1}{x+y}$ for some code I am writing. Using WolframAlpha to quickly compute the Maclaurin series gives me: $$ \frac{1}{y} - \...
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3answers
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Find Taylor series of $\sqrt{x}$ centered at $x=4$ and the order 3

Find Taylor series of $\sqrt x$, about $x=4$ and the order 3 I've tried a few timesm but I keep getting a result that does not comply with the answer. Following are the steps I've taken, hopefully I ...
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2answers
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$\frac{ \sin\theta }{ \theta } = \frac{2165}{2166}$ Find the approximate value of $\theta$

$\dfrac{ \sin\theta }{ \theta }$ = $\dfrac{2165}{2166}$ Find the approximate value of $\theta$ What is the method to solve this question. (I have tried solving it by using Taylor series expansion, ...
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What intrinsic property determines whether a function is analytic

Given we know the value of all order derivatives at a point $x_0$ for a given f(x). As per my knowledge all the geometric properties like slope, curvature, convexity are functions of solely the ...
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Finding Taylor series third degree polynomial. What form does the general equation for the term have to be in?

I am trying to figure out the Taylor polynomial of degree $3$, denoted as $T_3(x)$, for $f(x) = xe^{-2x}$. I am a bit confused about what form the general term of the series needs to be in for me to ...
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By creating a Maclaurin series up to an including $x^4$ for $\ln(\cos x)$ shows that $\ln2 \approx \frac{\pi^2}{16}\left( 1+\frac{\pi^2}{96}\right)$

By creating a Maclaurin series up to an including $x^4$ for $\ln(\cos x)$ shows that $$\ln2 \approx \frac{\pi^2}{16}\left( 1+\frac{\pi^2}{96}\right)$$ So creating a Maclaurin series using the general ...
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42 views

find taylor series but the center is not analytic

I want to find a Taylor Series for complex function $$f(z)=\dfrac{z^2}{2+z},$$ centered at $z=-2$. I have find the taylor series $f(z)$ centered at $z=z_0$, $$f(z)=(z-2)+\dfrac{4}{z+2}=(z-2)+4\sum\...
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How can Taylor series diverge?

I understand how to prove a power series diverges, but that seems to contradict simple logic for me. The idea behind constructing a Taylor series is that it is a polynomial that has the same nth ...
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Simplifying a marginal likelihood function

I have the following likelihood function $$ p( z \big| x, \lambda, \sigma) = \frac{1}{\sigma^{2}} \cdot \exp \bigg( -\frac{\big( z^{2} + \lambda^{2} \cdot x^{2} \big)}{ \sigma^{2} } \bigg) \cdot I_{0}...
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How many solutions does equation $\int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0$ have on the segment [0, 3]?

The task i'm trying to solve is: How many solutions (roots) does equation have: $$\int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0$$ on the segment [0, 3] ? By the moment i'...
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182 views

What does $F'$ and $F''$ mean?

I'm trying to learn what a Taylor series is, This is the equation I'm looking at and I know 0 calculus. I have been told that $F'(x)$ is a derivative but what does $F''(x)$ mean?
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How do I formally show the radius of convergence of the Taylor series of $f(x)=x^6 - x^4 + 2$ at $a=-2$?

This is an exercise in Stewart's Calculus (Exercise 19, Section 11.10 Taylor and Maclaurin Series): Find the Taylor series for $f(x)$ centered at the given value of a. [Assume that f has a power ...
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Taylor series of complex function confusion with big O notation

Suppose $u(x,t)$ is a function of two real numbers that outputs a complex number. Usually I would have $u(x,t+k) = u(x,t) + k\frac{\partial u}{\partial t}(x,t) + \frac{1}{2}k^2 \frac{\partial ^2 u}{\...
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2answers
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Find the Taylor Polynomial $T_{3}$ for the Function $f(x) = \frac{5x}{2+4x}$

Find the Taylor Polynomial $T_{3}$ for the Function $f(x) = \frac{5x}{2+4x}$ So I have this problem and I'm struggling, but below is what I am attempting to do: Plan: Attempt to translate series ...
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What is the Maclaurin series representation of $(1 - \frac{x}{5})^{-4}$

Question: Find the Maclaurin series representation of $(1 - \frac{x}{5})^{-4}$ using the definition of the Maclaurin series $\sum_{n=0}^{\infty} \frac{F^{n}(a)}{n!}(x-a)^n$ My approach: Find $F^n(0)...
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Find the Sum of the Series: $\sum_{0}^{\infty}\frac{(-1)^n\pi^{2n}}{6^{2n}(2n)!}$

Find the Sum of the Series $$\sum_{n=0}^{\infty}\frac{(-1)^n\pi^{2n}}{6^{2n}(2n)!}$$ Alright, so I think I may have gotten this problem correct but I'm a little hesitant, so If you could check my ...
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2answers
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Taylor series, the n-term calculation

Good afternoon. I need to claculate the general n-term Taylor's expansion at zero of 2 functions: $e^{-x^2}$ $e^{-\frac{1}{x^2}}$ if $x \ne 0$ and $0$ otherwise For the first function everything ...
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A question about Taylor formula [closed]

Suppose $f(x)$ is twice differentiable, $|f(x)|\le a$ and $|f''(x)|\le b$. For an arbitrary $c\in (0,1)$, prove $|f'(c)|\le 2a+b/2$.
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Using series to approximate definite integrals.

Use series to approximate the value of $\int_0^1{cos(x^2)dx}$ so that the error in your approximation is less than $\frac{1}{100}$. My work: $$f(x)-T_n(x)=R_n$$ $$R_n=|\frac{f^n (c) x^n}{n!}|$$ $f^...
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understanding the difference between Laurent and Taylor series.

In my homework, I have a problem that says, Set $f(z)$ = $\frac{e^{z^2}}{z^4}$. $(a):$ Find the Laurent series for $f$ centered at $z_0 = 0$ $(b):$ Let $C$ be the positively oriented unit circle. ...
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What is the radius of convergence of $i + \dfrac{2}{1-i}\displaystyle\sum_{j=1}^{\infty} \biggl(\dfrac{z-i}{1-i}\biggr)^{j}$?

When I apply the ratio test, I get: $\displaystyle\lim_{j \rightarrow \infty}\biggl| \dfrac{2(z-i)^{j+1}(1-i)^{j+1}}{(1-i)^{j+2}(z-i)^{j}}\biggr|$=$\displaystyle\lim_{j \rightarrow \infty} \biggl|\...
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Integral limits for taylor expansion

Imagine the following function,where $a,b$ are positive constants \begin{equation} f(x) = g(x + a),x \in [-a, b] \end{equation} Expanding $f(x)$ in its Taylor series, around $x$, we get: \begin{...
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Determining the McLaurin series of $(1-z)e^{-z}$

Determine the Mclaurin series and the convergence radius for $(1-z)e^{-z}$. I know the Taylor expansion of $e^t=\sum_\limits{n=0}^{\infty}\frac{t^n}{n!}$.Replacing $t=-z$ then I have $e^{-z}=\sum_\...
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Find the Taylor series of $\frac{1+z}{1-z}$ at $z_{0}=i$

I'm given the following explanation: Let $\dfrac{1+z}{1-z} = \biggl(\dfrac{1+i}{1-i}+\dfrac{z-i}{1-i}\biggr)\biggl(1-\dfrac{z-1}{1-i}\biggr)^{-1}$ = = $\dfrac{1+i}{1-i}\displaystyle\sum_{j=0}^{\...
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Summation Formula for Tangent/Secant Numbers

I came across the following expressions: $$\begin{align} \widehat{S}_{2n} &:= \sum_{1 \leq k_1<\cdots<k_n \leq 2n} \prod_{\ell=1}^n (k_\ell-2\ell)^2, \\ \widehat{T}_{2n+1}&:=\sum_{1 \...
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Determing Mclaurin series for $\sqrt{1-z^2}$.

Determine the Mclaurin series and the convergence radius for $\sqrt{1-z^2}$. I tried to adapt the series $(1+z)^\alpha=\sum_\limits{n=1}^{\infty} {{\alpha}\choose {n}}z^n$. $\sqrt{1-z^2}=1+\sum_\...
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Another version of Taylor expansion remainder in integral form

Most material I came across give the following standard form of Taylor expansion with integral remainder: $f(y)=f(x)+(y-x)f'(x)+\frac{1}{2}(y-x)^2 f''(x)+\int_x^y \frac{f'''(u)}{2}(y-u)^2du$ But I ...
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2answers
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Evaluating $\sum_{k=1}^{\infty}\frac{1}{k(3k-1)}$

I am wondering if the sum $$S=\sum_{k=1}^{\infty}\frac{1}{k(3k-1)}$$ has an exact expression. And when I plugged it into Wolfram Alpha it spitted out: $$S=\frac{1}{6}\Big(-\sqrt{3} π + 9 \ln(3)\Big)$$...
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What happens to the negative sign on $T_{m}(z)$?

We seek an expression for $\displaystyle 1/(\zeta -z)$ in powers of $\displaystyle(\zeta-z_0)/(z-z_0)$, whose magnitude is less than $1$; accordingly we write $\dfrac{1}{(\zeta-z)}=\dfrac{1}{(\...
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Taylor expansion for Logistic function near a point different than zero

How to derive Taylor series expansion for Logistic function near a point $x=a\neq0$ of the form: $$L(x) = \frac{1}{1+e^{-k(x-x_0)}}$$ Edit Using the general form for Taylor series coefficients $c_n=...
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1answer
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Estimating value and error with Taylor polynomial

I am trying to estimate $ln(\frac{3}{2})$ to three decimal places. I was trying to use Taylor series for $ln(x+1)$ with Lagrange's form of reminder. As $$(ln(x+1))^{(n)} = \frac{(-1)^{n+1}(n-1)!}{(x+...
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Are $C^\infty$ Functions with all derivatives positive on [a,$\infty$),a$\gt$0 always made of exponential?

Are there any $C^\infty$ real functions except the exponential family and gamma function family which has all the derivatives of same sign on an interval [a,$\infty$) with a$\gt$0 ? I speculate the ...
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Show that $\int_{S_r}{y_{i}y_{j}d\sigma(y)}=0$ on the sphere $S_{r}(x)$.

Let $S_r(x)$ the sphere of radius $r>0$ centered at the point $x\in\mathbb{R}^{n}$, that is $$S_{r}(x)=\{y\in\mathbb{R}^n : |x −y| = r\} $$ Let $\sigma$ be the $(n-1)$-volume on $S_r(x)$, and ...
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4answers
79 views

Prove $\sin(x) > x - \frac{x^3}{3!} $ on $(0, \sqrt{20})$

I'm having a bit of trouble with this because my attempted proof breaks down. Proof: It is sufficient to show that $f(x) = \sin(x) - x + \frac{x^3}{3!} > 0$ on $I = (0, \sqrt{20})$. This is true ...
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2answers
67 views

Find $f^{(22)}(0)$ for $f(x)=\frac{x}{x^{3}-x^{2}+2x-2}$

Find $f^{(22)}(0)$ for $f(x)=\frac{x}{x^{3}-x^{2}+2x-2}$ I know that I should use Taylor's theorem and create power series. However I don't have idea how I can find $a_{n}$ such that $f(x)=\sum_{n=1}^...
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57 views

Taylor series implying $f(-z)=0$ whenever $f(z)=0$.

Let $f:\mathbb{C}\mapsto\mathbb{C}$ be holomorphic, then $$f(z)=\sum\limits_{r=0}^{\infty}\frac{f^{(r)}(0)}{r!}z^r.$$ Writing $z\equiv|z|(\cos(\arg(z))+i\sin(\arg(z)))$, and using De Moivre's theorem, ...
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1answer
37 views

Determing the Mclaurin series of $\frac{z^2}{(1+z)^2}$

Determine the Mclaurin series and the radius of convergence for the function: $$\frac{z^2}{(1+z)^2}$$ I guess I am supposed to adapt the following series $\frac{1}{1+z}=\sum_\limits{n=0}^{\infty}...
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1answer
31 views

Expand $f(z)=\frac{1}{z(z-3)}$ as laurent series in domain $1 < |z-4| < 4$

Expand $f(z)=\frac{1}{z(z-3)}$ as laurent series in domain $1 < |z-4| < 4$ Any suggestion i have $\frac{1}{z-3}= \sum_{n=0}^{\infty} (-1)^n \frac{1}{(z-4)^{n+1}}$ and $\frac{4}{z}= \...
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30 views

Can someone help me expand this using Taylor's expression with Higher order terms?

I want to study this, for those who are curious this equation comes from optical flow brightness constancy assumption, but I could not find in the literature anywhere higher order terms for this ...
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20 views

Taylor Polynomial for Log(1-t) derivation {confused}

I need some help understanding something from my textbook. In the book they are deriving a taylor polynomial approximation for log(1-t), the first thing they do is integrate log(1-t) from 0 to t. ...