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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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Taylor coefficients.

I have a function $$f(x) = e^{-x(2a-x)}$$ It is easy to write out the Taylor expansion for this function in $x=0$ point. But.. I want to get a formula for the Taylor coefficients. I tried a lot. I ...
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43 views

How can I compute the nth derivative of a function without writing by hand [on hold]

How would I compute the 6th derivative of $\frac{\cos{(5x^2)}-1}{x^2}$?
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Does $\sum^\infty_{j=a}{\frac{x^{j+(j+b)}}{j!(j+b)!}}$ converge to some function?

I was wondering if $\sum^\infty_{j=a}{\frac{x^{j+(j+b)}}{j!(j+b)!}}$ converges to some function just like the series expansion of $\exp(x)$. As for the series, $a$ and $b$ are arbitrary non-negative ...
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39 views

Probability mass function from a generating function

I have the generating function $G_x(\theta) = \frac{\alpha-1}{\alpha-\theta^2}$ and I am trying to determine the probability mass function. I believe I need to determine the Taylor series expansion ...
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4answers
61 views

How to “directly” obtain Maclaurin series of $\exp(x-1+\sqrt{x^2+1})$

Consider the Taylor expansion centered around $x_0 = 0$ for $$f(x) = e^{x-1 + \sqrt{x^2+1}} ~,\qquad x\in\mathbb{R}$$ The goal is to arrive at $\sum c_k x^k$ by hand, and I wonder if there is an ...
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1answer
35 views

Property of alternating sign series

I have an alternating series $\sum_{i=0}^{\infty}(-1)^{i+1}a_ix^i$, with $a_i\geq 0$ and the series is easy to check to converge for any $x>0$. I numerically checked that this sum is negative for ...
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57 views

Find $x+ \frac {2}{1 .3} x^3 + \frac {2.4}{1.3.5} x^5 + \dots \infty $ for all $x \in (0, 1)$ [duplicate]

Find the closed form of $$x+ \frac {2}{1\cdot 3} x^3 + \frac {2\cdot 4}{1\cdot 3\cdot 5} x^5 + \dots \quad \forall x \in (0, 1)$$ My approach: Clearly we can see the formation of factorial in the ...
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33 views

Multivariable Calculus - Approximation

I need to show that for any approximation around the point (0, 0) is valid: $$\lim _ { x \rightarrow 0 , y \rightarrow 0 } \frac { \sin ( x y ) } { x y } = 1$$ I know how to do Taylor series ...
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14 views

How to take derivative of multivariate Taylor series matrices?

Suppose we consider a function $f(x)$ with $f(x):\mathbb{R}^n \to \mathbb{R}$. We let $r(x)$ be the second order Taylor series of $f(x)$ about the base point $z \in \mathbb{R}^n$. How can I show ...
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19 views

Error expansion for Trapezoidal rule

From my lecture slides, the error expansion for the trapezoidal rule was stated as (even powers of h only) but from the Taylor series expansion of the exact and trapezoidal rule I can't ...
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1answer
31 views

Small-Angle Approximation for Cosine

The small-angle approximation for cosine is: $$ \cos (x) = 1 - \frac{x^2}{2} $$ Question: How can I find a range of values of $x$ for which this approximation gives correct results rounded to 2 ...
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1answer
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Finding Taylor series for a complex logarithm branch

The Problem: Find the Taylor series for the logarithm branch $0<\arg(z)<2 π$ in powers of $z+2$ My resolution: The method I've used to come up with an answer was integration of a known series. ...
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1answer
51 views

$\cos x + \cos 2x = 1$. Maclaurin series.

I am required to approximate $\cos (x) + \cos (2x) = 1$ using the first three non-zero terms in the respective Maclaurin series. I have found the first three non-zero terms for both $\cos (x)$ and $\...
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2answers
534 views

Applying the induction hypothesis indirectly? I have trouble understanding this proof.

I am confused about the mechanics of the following proof (page 423, chapter 20, of the fourth edition of Spivak's Calculus (Taylor's Theorem)): I am not sure that I understand how applying the ...
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1answer
20 views

Perturbation expansion within trig function

I'm trying to find an approximate solution to a nonlinear differential equation. It involves something to the effect of $\frac{d\Psi}{ds} = \sin{\Psi} + \dots$ , where $\Psi$ is a small variable. If ...
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1answer
26 views

How to prove the variance of a function g(X) of the random variable X is given by $var[g(X)]\approx\{\frac{dg(X)}{dX}\}^2var(X)$

I am reading the book Modelling Survival Data in Medical Research. The book says that it's the Taylor series approximation to the variance of a function of a random variable. I learnt Taylor expansion ...
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1answer
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Taylor/Maclaurin Series vs Taylor/Maclaurin Polynomial

For this question, I will use $sin(x)$ as an example. The Maclaurin series for $\sin(x)$ is $$\sum_{n=0}^{\infty} \frac{{(-1)}^nx^{2n+1}}{(2n+1)!}$$ This gives us a Maclaurin polynomial to represent $\...
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leading-order correction

What is a leading-order correction? The term is in the second part of a homework question I have and I do not understand what it means. The question is: a) Assuming that v/c << 1, find the ...
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Taylor expansion of an integral with respect a parameter

In a paper there is this integral: $$I=\int_{-\infty}^{\infty} \frac{dp}{2\pi} \ln (1+e^{-\frac{E(p)}{T}})$$ where $E(p)=2|sinh^{-1} sin(p/2)|=|p|+O(p^3)$ and $T>0$ ($T$ is temperature, $p$ is ...
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1answer
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Write the Taylor expansion of order $2$ at $x=0$ of $h(x)=g^{-1}(x+\sin(x))$, for $g(x)=x\ln(2+x^2)$

Can anyone tell me whether I carried out properly this exercise and where are mistakes? Thank you. Let be $g: \mathbb{R} \to \mathbb{R}$ the function defined by: $$g(x)\,=\,x\ln(2+x^2)$$ Show that $...
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How can I prove $\frac {d}{dx} {x^n} = n x^{n-1}$ for $ n \in \Bbb R$ without circular reasoning? [duplicate]

I just cannot prove that $$\frac {d}{dx} {x^n} = n x^{n-1}$$ for $ n \in \Bbb R$. For $n \in \Bbb{N}$, I can use the definition of a derivative : $$\frac {d}{dx}x^n = \lim_{h \rightarrow 0} \frac{(...
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Is there a way to compute the exponential of a PDP-1 matrix?

I am computing the exponential of a matrix via Taylor expansion to prove the end-result with induction. For a matrix $A=PDP^{-1}$ where $D$ is the diagonalized matrix, is there any kind of formula ...
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3answers
93 views

Constructing an odd trigonometric function with no 1st-order term in Taylor expansion

Is it possible to construct a polynomial $f$ of at least one of (the fewer, the better) $$\sin{x},\cos{x},\sin{y},\cos{y},\sin{z},\cos{z}$$ with the following properties? The smaller degree of $f$, ...
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2answers
31 views

Calculate limits with Taylor

I must calculate $$\lim_{t→1} \frac{\sin (t) − \sin (1)}{t − 1}$$ I can calculate it with L'Hospital's rule but I do not understand how to do it with taylor polynom. I know sin is : $$\sin x = \sum^...
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1answer
71 views

Taylor series over a sphere in $\mathbb{R}^3$.

I need some hint to prove this: Consider the subset $E\subset\mathbb{R}^3$ and the open ball $B_{R}:=\{(s,\theta); 0<s<R,\theta=(\theta_1,\theta_2)\}\subset E$, centered in $x\in E$. Let $y\in ...
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1answer
81 views

Estimating a Lebesgue integral and Taylor's formula

Suppose $V:\mathbb{R}^n\to \mathbb{R}$ is a $C^1$ function and let $DV$ the gradient of $V$. Let $A\subset \mathbb{R}^m$ compact and $f:\mathbb{R}^n\times A\to \mathbb{R}^n$ be a function such that $f$...
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Gradient descent expression help

In the section 2 of this paper. I didn't understand the following steps in the proof. If $x$ is channel input, $w$ channel weight vector and $\hat{w} = w/$ $ ||w||_2$ then for batchnorm ($BN$) we ...
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Find the fifth and tenth derivatives of a function using Taylor series

So,I found a problem where I have to evaluate the fifth derivative evaluated on $\ x=1 $, and the tenth derivative on $\ x=0 $ for the following function: $\ f(x)=(x^2-3x)e^{x^4} $ I am supposed to ...
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1answer
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Lagrange Remainder: open or closed interval?

As far as I can tell, if you start with the integral form of the remainder of a Taylor polynomial then you can derive the Lagrange form by an application of the mean value theorem for integrals. From ...
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Error approximation Taylor polynomial (multivariate)

Estimate the error in replacing $\cos{x}/\cos{y}$ by $\frac{1}{2} > \left(y^2-x^2\right)+1$ for $|x|,|y|<\pi/6$. Answer takes the fourth term in the Taylor expansion $\frac{1}{24} \left(x^4-6 ...
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3answers
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Evaluate $I = \int_{0}^{1} \frac{e^x - 1}{x} dx$ with Taylor Polynomial.

Evaluate $$I=\int_0^1 \frac{e^x-1}{x} dx.$$ The way I am trying to do this is by substituting $e^x$ with $p_n(x)$ and $R_n(x)$, the $n$-th Taylor Polynomial and its associated remainder. Would it be ...
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42 views

Calculate the series $1/(k+2)k!$

Let $S = \sum_{k=0} ^\infty \frac{1}{(k+2)k!}$. I am trying to evaluate this sum. I tried using the Taylor series of $e^x$, which is similar, but I am not sure how to deal with the $1/k+2$ factor. Any ...
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Taylor's Polynomials and Derivative Test

I'm so confused about the Taylor's Polynomial and Series (including approximation of a function using Taylor's Series) and derivative tests and come up with a few questions. So, we approximate a ...
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3answers
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Prove $e^x$ is its own derivative via power series?

$$ \frac{d}{dx}e^x =\frac{d}{dx} \sum_{n=0}^{ \infty} \frac{x^n}{n!}$$ $$ \sum_{n=0}^{ \infty} \frac{nx^{n-1}}{n!}$$ $$ \sum_{n=0}^{ \infty} \frac{x^{n-1}}{(n-1)!}$$ This isn't as straightforward as I ...
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1answer
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Can you convert a taylor series into a power series while avoiding singularities/discontinuities that result from a=0?

Suppose one follows the Taylor series formula for a function, defining a function as $$ \sum_{n=0}^{ \infty} \frac{f^{n}(x)|_{x=a}}{n!}(x-a)^n.$$ Then, suppose you want this function to instead be ...
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Inverse matrix series expansion

I have a matrix $$ M(\lambda) = I-\lambda A+B\sum_n \lambda^n e_n $$ where $\lambda$ is a scalar variable, $e_n$ is a series of known scalar, $I$ is an $n\times n$ identity matrix, while $A$ and $B$...
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Exponential of the product between $x$ the derivative operator of $x$ acting in a $f(x)$

The question I'm stuck here trying to figure out how to compute and prove, the following operator action in a function: $\exp(\varepsilon x \partial_x) f(x) = f(x \exp(\varepsilon) )$ where $\...
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1answer
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Proof of $|\sin(x)| \leq 1$ and $|\cos(x)| \leq 1$

Using the series representations of $\sin(x)$ and $\cos(x)$, how does one show that both $|\sin(x)| \leq 1$ and $|\cos(x)| \leq 1$? I can do this easily algebraically/trigonometrically, but I am ...
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1answer
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How do I show $\sum\limits_{j=0}^{\infty}e^{-wj}=\frac{1}{w}+\frac{1}{2}+\frac{w}{12}-\frac{w^3}{720}+\frac{w^5}{30240}+\cdots$?

Since $$\sum_{k=0}^{\infty}x^k=\frac{1}{1-x},$$we have $$\sum_{j=0}^{\infty}e^{-wj}=\sum_{j=0}^{\infty} (e^{-w})^j=\frac{1}{1-e^{-w}}=\frac{e^w}{e^w-1}$$ Also, since $$e^x=\sum_{n=0}^{\infty}\frac{x^...
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Why is this series given by this Taylor expansion?

So I am doing some graph theory stuff and I know that, given $4$ axes that I can walk along in both directions, the number of reached points after $n$ steps is: $$ N(n) = \sum_{k=0}^{\mathrm{min}(4,n)...
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1answer
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Simplifictaion: Taylor series expansion $U(x,t)$ about point $\bigg(x+\frac{1}{2}h , t\bigg)$

I am required to expand the $\partial_x^2U(x,y)$ in terms of a finite difference expression about the point (x+\frac{h}{2},t) instead of the usual $(x,t)$ point. This means one will get: $$ \partial_x^...
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28 views

Taylor series for with an integral

I was trying to analyze a large amplitude problem and I got stuck at an equation like this. $$\int_0^T dt=\sqrt{\frac{l}{2g}}\int_{\theta_{\text{max}}}^{ \theta_0} \frac{d \theta }{ \sqrt{\cos\theta -...
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Can I “naïvely” claim that (complex) Taylor series are all analytic because…

Can I "naïvely" claim that (complex) Taylor series are all analytic because each one of their terms is clearly entire? The reason for my hesitation is because there are infinitely many terms ...
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1answer
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Find an upper bound for $|f(x)-P_4(x)|$, for $0 \le x \le 0.4$

So I'm stuck on something that is supposed to be fairly easy... I was able to work through the Taylor polynomial which I believe to be: $$x+x^3+\frac{x^5}{2!}+\frac{x^7}{3!}+\frac{x^9}{4!}$$ But I'...
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1answer
41 views

Evaluate the sign of an integration

Let $P_k(x)$ to be the first $k+1$ terms of the Taylor expansion of $\cos(x)$, that is $$P_k(x) = \sum_{l = 0}^k (-1)^l x^{2l}/(2l)!.$$ For $\alpha>0$ and $\alpha\notin \mathbb{Z}$, I want to ...
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2answers
31 views

How to find the power series expansion that converges to Fresnel integral?

Fresnel integral is $S(x)=\int_{0}^x{\sin(t^2)\,dt}$. I'm trying to see how the power series expansion for the integral is found , I have to tried to use Taylor Series for expanding $\sin(t^2)$ but i ...
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2answers
36 views

Does the existence of Taylor polynomial of order n imply $C^n$?

Let's say that I want to prove that the function $$f(x)=\left\{\begin{array}{cc} e^{-\frac{1}{x^2}} & x\neq 0 \\ 0 & x=0 \end{array}\right.$$ is $C^\infty$. I did it using calculation by ...
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16 views

Expansion in power of $\frac{1}{Z}$ and $\frac{ln(Z)}{Z}$

When I read the paper I met the problem in the step expansion in power. We have \begin{align} s(\epsilon)=\frac{A\epsilon^{a}}{bB|\dot\epsilon|}e^{-Be^{b}} \left[1+\frac{a}{bB}\epsilon^{-b}+\frac{a(a-...
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12 views

Do these integrals have a simple expression in terms of local function properties?

Recall the following identity, assuming continuity of $f$: $$\lim_{\Delta x\to0} \frac{\int_{x_0}^{x_0+\Delta x}f(x)}{\Delta x}=f(x_0)$$ Can similar expressions in terms of $f(x_0)$ or its ...
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1answer
12 views

Function representation of power series convergent at x = a

SOME BACKGROUND INFO: Analytic functions may be (locally) represented by a convergent power/Taylor series. The domain is given by the interval in which the power series represents this function. For ...