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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26 views

Proving two polynomials converge to the same function

I am looking to create polynomials that converge past the usual radius of convergence on the real line. So far, I have proved that this polynomial will converge to the analytic continuation of the ...
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1answer
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Find the radius of convergence of a polynomial with limited terms

The complete question is to find the Taylor series for $f(x)=x^5+3x^3+x$,centered at $a=3$ and then to find the associated radius of convergence R. The Taylor series is straight forward and I get: $...
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Question concerning Taylor series of $\sin(x)$ and Lagrange's remainder theorem

Consider the Taylor polynomial of 5th degree of $\sin(x)$ centered at zero: $$ T_5(x) = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!}. $$ Lagrange's remainder theorem gives us an expression for the remainder $...
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35 views

Taylor Series representation of $f(x) = \sqrt{x} + \frac{1}{\sqrt x}$ at $a=1$

I am trying to find the Taylor Series representation of $f(x)= \sqrt x + \frac1{\sqrt x}$ at $a = 1.$ With $5$ terms. I know how to get the series expansion. centered at $a=1$. with $5$ terms… However ...
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What are first order residual, second order residual, third order residual…

What are first-order residual, second-order residual, third-order residual...and so on. Like the difference between the original function (evaluated at some value) and the approximated function (to ...
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What is $\lim \limits_{n\to\infty }\frac{n^x}{n!},\ n\in \mathbb Z^+,\ x\in \mathbb R$? Does $n^x$ or $n!$ grow faster as $n\to \infty $?

The question just pops up in my head. $n^x$ is just an exponential function with positive integer base. My intuition is that $n^x$ grows faster so the limit should be infinity. How do we properly ...
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59 views

Is taking a Taylor expansion of $x^{x^x}$ possible?

Disclaimer: I do not know that much math, so there might be obvious mistakes that I make in this question. As far as I can tell, taking the derivative of $x^{x^x}$ infinitely is possible. However, ...
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show that x-x^(3)/6 < sinx< x for x>0. ( taylor series)

the question in a random study problem says show that x - $\frac{x^3}{6} < sinx< x $ for $x>0$. ( taylor series). I am assuming the question means about x=0 I have tried finding the maximum ...
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19 views

Finding the norm of the Taylor approximation of a multivariate function $f:\mathbb{R}^n\mapsto\mathbb{R}^m$

I have a function $f:\mathbb{R}^n\mapsto\mathbb{R}^m$. My goal is to bound the first order Taylor approximation of $f$. Given $x,x'\in\mathbb{R}^n$ I have that \begin{equation} f(x)-f(x')\approx (x-x')...
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Not sure where Term comes from in Expansion of Geometric Sequence

Quick question - I'm just wondering how that $\mu_1$ term appeared in the second equality next to the $x_1$ term.
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Functions as Taylor coefficients

I'm stuck at the following lemma. Let $f:[a,b]\to\mathbb{R}$, and suppose that there exists $\phi_1,...,\phi_n:[a,b]\to\mathbb{R}$ continuous functiosn such that: \begin{equation} f(x+h) = f(x) + \...
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How does one treat big O when Taylor expanding $\sin(\sin x)$?

I am working on finding the Taylor (Maclaurin expansion) of $\sin(\sin(x))$ to the third order. We have $$\sin{t} = t - \frac{t^3}{6} + \mathcal{O}(t^5)$$ If I then set $t = \sin(x)$, and expand once ...
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Show that $x-x^3/6<\sin x<x$ for $x>0$ [closed]

the question in a random study problem says show that $x-x^3/6<\sin x<x$ for $x>0$. (taylor series). I am assuming the question means about $x=0$ I have tried finding the maximum error ...
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35 views

Find the Taylor series of $(1-x^2)/(4-x)$ [closed]

i got this question on my test and so far I have no idea how I should use approach that problem. hope some of you could help me
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20 views

Uniform Convergence of Remainder exponential function

For $x \neq 0$, define $f(x)=\exp \left(-1 / x^{2}\right).$ Let $f(0) = 0$. Show that the sequence of Taylor polynomials of $f$ at $x_{0}=0$ cannot possibly converge to $f$ except at the one point $x_{...
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22 views

Asymptotic behaviour of coefficients of converging power series

Suppose $\sum_{n=0}^{\infty}a_n z^n$ is a power series whose radius of convergence is $R=1$ (this can be generalised by scaling). What can we say about the coefficients $\{a_n\}$ of the power series? ...
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1answer
45 views

Let $P(x)=a_0+a_1x + a_2x^2+..+a_nx^n$. $a_n\neq0$ $[1]$

Let $P(x)=a_0+a_1x + a_2x^2+..+a_nx^n$. $a_n\neq0$ $[1]$ I have that Now I want this Taylor polynomial to be $x-x_0$ instead of $x$ degrees. With binomial formula we have $x^m=[(x-x_0)+x_0]^m=\...
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Uniform Convergence of Taylor Remainder

For $x$ in $\mathbb{R}$, let $f(x)=(x+1) e^{x}$. After finding the $k$ th Taylor polynomial for $f$ at $x_{0}=0$ and the corresponding Lagrange form of the remainder $R_{k}(0 ; x)$, prove that $$\lim ...
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How do I find Maclaurin-like series for parametrically defined curves?

Suppose I have a parametrically defined curve: $$x=x(t),\quad y=y(t).$$ For some parametrically defined curves, such as $x=t^2,y=\sin(t^2)$ (a very basic example, I know) we can eliminate $t$ and ...
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Taylor polynomials' plots begin to look odd in graphing software for high-degree polynomials

I was playing around with Taylor approximations using some graphing software. I am using $-\sin\left(x\right)\approx-x+\frac{1}{3!} x^{3}-\frac{1}{5!} x^{5}+ \ldots$. When I turn the Taylor polynomial ...
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Taylor's Theorem and natural logarithmic inequality

Use Taylor's theorem to prove that, for $x>0$. $$ \ln x+\frac{1}{x}-\frac{1}{2 x^{2}}<\ln (x+1)<\ln x+\frac{1}{x} $$ The RHS is indeed obvious: algebraic manipulations yield to show $e^{\frac{...
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Finding the max value given table

So I was given the following prompt: Let $f$ be a function having derivatives for all orders of real numbers. The function and its first four derivatives at $x=0$ are given in the table below. The $5$...
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Showing the existence of a Taylor Series with convergence as a prerequisite.

I had this issue understanding how to start with the following problem that I have found in a book involving the Taylor Series. The problem goes as follows: Let $f(x) = \sum_{k=0}^\infty a_k(x-x_0)^k$...
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Finding remainder of a Maclaurin Series

So I was given the following prompt: "The function $f$ has derivatives of all orders for all real numbers and $f^4(x)=e^{2x-1}$. If the $3$rd degree Taylor polynomial for $f$ about $x = 0$ is ...
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Aproximation in the central difference formula

I need help to solve the following problem: enter image description here The exercise suggests following the idea of the proof of Lemma 2.4.2 which says the following: enter image description here ...
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1answer
22 views

Using maclaurin expansion find the limit $\lim e^{1/x}(x^3-x^2+x/2)-(x^3+x^6)^{0.5}$ as x approaches infinity.

I tried to write - $$e^x = 1+x+x^2/2+x^3/6+\alpha(x)x^3$$ $$e^{1/x}=1+1/x + 1/2x^2 + 1/6x^3 + \alpha (1/x)/x^3$$ That did not help
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Taylor series expansion of function of a binomial random variable.

I know that one can approximate the $E(X\times f(X,Y))$ by using the Taylor expansion of $X \times f(X,Y)$ when X and Y are continuous random variables. However, does the method work if $X$ and $Y$ ...
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2answers
37 views

Zeros of $f:R^2 \rightarrow R$ with Newton-Raphson?

I would like to find zeros of $f:R^2\rightarrow R$ applying the Newton-Raphson method but I got stuck in solving the linear approximation equation for $\textbf{x}$. Let $\textbf{x}\equiv(x,y)$ and $\...
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29 views

Series method to solve ODE with initial condition

Using $$y(t) := \sum^{ \infty} _{k = 0} a_kt^k$$ to solve $$y'' - t^2y' = \ln(1+3t)$$ with y(0) = 1 and y'(0) = 0 I first expand $ln(1+3t)$ using Taylor series at t = 0: $$\sum_{k=0}^{\infty}(-1)^{k-1}...
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Why does the Maclaurin serie of $(1+x)^k$ converge only for $|x|< 1$?

So I understand that if $f(x)=(1+x)^k$ you get something like \begin{align} f(x)=(1+x)^k \Rightarrow f(0)=1\\ f'(x)=k(1+x)^k \Rightarrow f'(0)=k\\ \vdots\\ f^{(n)}(x)=k(k-1)\dots (k-n+1)x^{k-n} \...
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153 views

Determine the limit using Taylor expansions:

I struggle with this one—maybe someone could point me in the right direction. $$\lim_{x\to 0} \frac{5^{(1+\tan^2x)} -5}{1-\cos^2x}$$ Getting the Taylor series expansion for $\tan^2x$ and $\sin^2x$ is ...
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Is it useful to expand the likelihood in Taylor serie *without using* the log likelihood ? Does it make sense?

The log likelihood has the advantage of transforming the product in sum. The likelihood is often expanded in a Taylor serie around the best solution, by using the log likelihood in the Taylor ...
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1answer
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Taylor expansion of $\frac {1}{|x-y|}$with x and y two vectors

This equation comes from a physics script on electrodynamics, saying that this equation comes from a Taylor series expansion. I understand the first equality, but not the second one. It is really not ...
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How to solve this system of equations involving factorials

I've encountered the following problem while doing research (trying to invert a Taylor expansion). Let $$ c_{\alpha} = \begin{cases} \frac{1}{k!}, &\mbox{if } \alpha = k e_k = (0,\dots,0,k,0,\dots,...
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79 views

Asymptotic behavior and a function less than $x$

Inequality found with the help of Desmos and WA: Let $x>0$ sufficiently large then we have : $$x^{\frac{x}{x+1}}-\left(\frac{x}{x+1}\right)^{x}+\ln\left(x\right)<x<x^{\frac{x}{x+1}}-\left(\...
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Converse of Taylor expansion theorem

The Taylor theorem states, that if a function $f$ is $n$-times differentiable at point $a$, it has a unique expansion $$f(x) = \sum_{k=0}^n f^{(k)}(a) \frac{(x-a)^k}{k!} + o((x-a)^n)$$ Suppose now we ...
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Is the $\xi_n$ of the Lagrange's remainder of $\log(1+x)$ monotonical?

This question is a special case of my previous one, Are there any $f(x)$ whose $\xi_n$ of the Lagrange's remainder does not converge to $0$?. Consider the Maclaurin's series of $\log(1+x)$ with ...
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Can you Taylor series a Step Function? Or distributions in general?

Is there a sense in which one can "Taylor Series'' a distribution? For example, is it true that $$ \Theta(x - \epsilon) = \Theta(x) - \epsilon \delta(x) + \frac{\epsilon^2}{2} \delta'(x) + \ldots ...
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1answer
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Taylor series of $2^\frac{x_\text{ideal}-x}{d}$

I have a question about Taylor series I need to compute Taylor series of this: $2^\frac{x_\text{ideal}-x}{d}$ The f'(x) is: $-\dfrac{\ln\left(2\right){\cdot}2^\frac{x_\text{ideal}-x}{d}}{d}$ The f$''$(...
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Modification of Lagrange’s Remainder Theorem to calculate $\ln(2)$

The following question is from Stephen Abbott's "Understanding Analysis." Question: Explain how Lagrange’s Remainder Theorem can be modified to prove $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4} \...
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Taylor series expansion of $f(x_n + f(x_n))$

Currently I am stuck at trying to understand a proof that uses the fact that the Taylor series expansion of a continuous function $f$ at $f(x_n + f(x_n))$ is $f(x_n + f(x_n)) \approx f(x_n) + f'(x_n)f(...
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Estimating the Lagrange Remainder in an interval

Let $f(x) = \frac{1}{1-3x}$. Determine the Taylor Series at $x_0=7$ and the Taylor polynomial of degree 2. Estimate the Lagrange remainder in the interval $[-4, 10]$. I started out determining the ...
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Extension of Taylor expansion of complex function

Suppose I have a complex function $g(z)$ defined on a disk $|z|\leq R$. If $g(z)$ is analytic on a smaller disk, say $|z|\leq R/2$, and thus have Taylor expansion $g(z)=\sum_{n=0}^{\infty}c_n z^n$ on $...
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1answer
61 views

Taylor Expansion for $\ln\sqrt{x^2-2x+2}$ at $x_0=1$. [closed]

I have just started to learn Taylor expansions. I am confused in this question. Can you explain its Taylor Expansion at $x_0=1$ clearly? $\ln\sqrt{x^2-2x+2}$ at $x_0=1$. Thanks.
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1answer
40 views

Taylor expansion for $\frac{x-1}{x^2-2x+2}$ at $x_0=1$.

I have just started to learn Taylor expansions. I am confused in this question. Can you explain its Taylor Expansion at $x_0 = 1$ clearly? $$f(x) = \frac{x-1}{x^2-2x+2}.$$
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2answers
34 views

Approximating using Taylor series expansion

Using linearisation (truncated Taylor series expansion) around 0, show that: $1-2 \omega \sin ^{2}\left(\frac{k \pi h}{2}\right)$ $\approx$ $1-\frac{\omega k^{2} \pi^{2}}{2} h^{2}$ I am a bit lost on ...
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50 views

Finding the best Taylor-esque polynomial

Background Based only on the information provided by derivatives at a single point, the Taylor series provides the 'best'* approximation of the function around its radius of convergence. But, some ...
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2answers
38 views

Error term in asymptotic expansion of $(1+ax)^{-1/x}$ as $x\to\infty$

According to WA we have as $x\to\infty$ $$ (1+ax)^{-1/x}=1-\frac{\log ax}{x}+O(1/x^2). $$ I am confused on how one obtains the error term. Here is what I tried $$ (1+ax)^{-1/x}=\exp(-\tfrac{1}{x}\log(...
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1answer
14 views

Composition of two functions - Taylor series

Let $f$ and $g$ be $n$-times differentiable functions, and let us assume that the composition $ F (x) = f (g (x)) $ is well-defined on an interval. Let us say that the composition is also $n$ times ...
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1answer
53 views

Expansion of $e^x$ - correct form

I have come across in a textbook to an expansion of e to the x in the following form: $$ 1+ \frac1x + \frac1{x^2} + \frac1{x^3} + \ldots $$ Is the above correct or is it a typo? I am familiar with ...

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