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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33 views

How to arrive at this sufficient condition for convergence of Taylor Series?

So I have been studying the convergence of Taylor series from Tom M Apostol. There is a theorem which states the sufficient condition for convergence of Taylor series. Quoting the theorem we've ...
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Under what condition, a function can be upper bounded by its second-order Taylor expansion?

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$, and $g(x) = f(a) + \nabla f(a)^\top (x-a) + \frac{1}{2}(x-a)^\top \nabla^2f(a) (x-a)$ be its second-order expansion at $a$. We know that if $f$ is concave,...
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Laurent series of a cosine function

I want to find Laurent series of the complex function $$f(z) = \cos\left(\frac{z^2-4z}{(z-2)^2}\right)$$ at $z_0=2$. I will be thankful for any hints.
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Having problem finding an error that is less than what is given.

so I have a problem with a problem that has to do with Taylor polynomials. Here is what they want me to do: "Find a value do cos(1/10) with means of Taylor polynomial method, with an error less than ...
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How to prove relation between $(1-x)^k$ and $(1-x^k)$?

I have been trying to prove or disprove following inequality: $(1-x)^k \geq (1-x^k)$ $\forall 0 \leq x \leq 1 $ and $k \in \mathbb{N}$. I thought about taking $\log$ on both sides and comparing but ...
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4answers
66 views

Taylor series for exp(exp(x)) using just the power series for exp(x) [duplicate]

I'm trying to figure out how to calculate the power series for exp(exp(x)) using exp(x) and then to write down the first few terms. I have the answer for the terms but I don't know how they arrived at ...
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Taylor in two variables - finite difference

I know that $U_{j\pm1}^n=u(x_{j\pm 1},t_n) \approx u\pm hu_x+\dfrac{h^2}{2!}u_{xx}\pm \dfrac{h^3}{3!}u_{xxx}+\mathcal{O}(h^4)$ and $U_{j}^{n+1}=u(x_{j},t_{n+1}) \approx u+ku_t+\dfrac{k^2}{2!}u_{...
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2answers
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Why Does Adding the nth Derivative Increase a Function Approximation's Accuracy?

I am currently taking calculus 3: sequences and series, and we've just started learning about Maclaurin and Taylor Series. I understand the concept behind them -- of these polynomials and derivatives ...
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Maclaurin series degree 6 involving substitution

So I am aware that you can perform operations on taylor series, such as integration, differentiation, etc. However, I am not sure of when exactly one is allowed to substitute values into another ...
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5answers
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Taylor Series - why that specific form?

Something that has bugged me since university. Why does the Taylor Series have that specific form? For example there is a division by n! - why not (say) (n^2)! How does one get to the Taylor Series? ...
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Matrix log Taylor expansion?

Given $\log(A + X)$ with $X\ll A$, is there a Taylor series like expansion one can do? That is, something of the form $$\log(A + X) = \log(A) + ...$$ I cannot assume $X$ and $A$ commute and in ...
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1answer
38 views

Taylor first order expansion for multivariable function using total derivative

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x,y) \rightarrow 3x - x^3 - 2y^2 - y^4$. Find the first order Taylor expansion $T^1_a$ in $a = (\frac{1}{2}, \frac{1}{2})$ The main problem is that ...
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How to compute newtonian gravitation from an infinite array of attractors?

In a flat toric universe (up connects down, right connects left and front connects back), every points repeats at $size_x$, $size_y$ and $size_z$ intervals. In such case the Newtonian gravitational ...
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Find a value of $\ln 1.2$ with accuracy of $10^{-4}$

I know the formula that helps find an approximate value. In this instance it would be like $\ln 1.2 = \ln (1 + 0.2) \approx 0 + 1 \cdot 0.2 = 0.2$. But I need to find the value more precisely. I ...
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How to prove Shannon entropy inequality with something that seems to be some sort of taylor expansion

I'm slightly confused about some sort of "proof" (probably not a real proof since it's physics math) I have the formula $f(x) = f(y) + (x-y)f'(y) + \frac{f''(y)}{2\epsilon}, \quad \epsilon \in (x,y)$...
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76 views

Convergences of Machin's Original Formula and Leibniz's Formula for $\pi$ [closed]

Does anyone know why Machin's original formula for $\pi$ converges so much faster than Leibniz's formula for $\pi$? Machin's original formula: $\pi=\sum_{n=0}^\infty \frac{16(-1)^n239^{2n+1}-4(-1)^n5^...
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Using Taylor Polynomial to Degree 5 to Approximate Machin's Formula for $\frac{\pi}{4}$

CONTEXT: Uni question made up by lecturer How would use the Taylor polynomial to degree 5 to approximate $4\arctan(\frac{1}{5})-\arctan(\frac{1}{239})$ Recall Machin's formula: $\frac{\pi}{4}=4\...
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Conjecture that certain series coefficient is always positive

I have the following conjecture: For non-negative integers $n$, $l$ satisfying $n \geq l+1 $, expand the function $$I_{n,l}(q,r) = ((1 + q)(1 + r))^{n + l} \int_0^1 dz\frac{z^{2 l + 2}}{(\frac{1}...
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Manipulation of Leibniz Series

Recall that the Leibniz series is $\frac{\pi}{4}=\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$. Can this be rearranged to give us $\pi=4\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}=\sum_{n=0}^\infty \frac{4(-1)^n}{...
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Maclaurin expansion of $\exp(x)/(1-\exp(x))^2$

I am trying to expand following function $f(x)=\frac{e^x}{(1-e^x)^2}$ using the Maclaurin series. According to wolframalpha it should be: $\frac{1}{x^2}-\frac{1}{12}+\frac{x^2}{240}-\frac{x^4}{6048}+...
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Is there a version of Taylor's theorem with integral remainder which is valid if the function is only differentiable in a weak sense?

Short question: Is there a version of Taylor's theorem with integral remainder which is valid if the function ($\mathbb R\to\mathbb R$) is only differentiable in a weak sense? Or at least if the ...
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Taylor Expansion of marginal value of capital

How to proceed with the log-linearization of the following equation: I tried the following: last two terms on RHS would not be taken into account. then take logs on both side and use taylor ...
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1answer
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Exercises combining Taylor polynomials and the Chain Rule

How can I approach this exercise using the Chain Rule? Let $f: \mathbb{R}^2 \to \mathbb{R}$, $C^2$ function whose second degree Taylor polynomial centered at (1,0) is $P(x,y) = 4 + x + xy + \frac{y^...
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A family of polynomials for the Taylor expansion of the Lambert-w function

This is an attempt to derive the Taylor expansion at $0$ of the Lambert $w$ function (the branch containing the origin of the solution $w$ to the functional equation $w(x)e^{w(x)}=x$) without ...
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4answers
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Finding the $n$th derivative of a function when $x=0$ using Maclaurin series.

CONTEXT: Uni question made up by lecturer So I have found the Maclaurin series of $f(x)=x^2e^{x^2}$ to be $\sum_{n=0}^\infty \frac{x^{2n+2}}{n!}$ which can be rewritten as $\sum_{n=0}^\infty x^{n+2}\...
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1answer
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If $f'$ is Lipschitz continuous with constant $c$, then $f(y)-f(x)-f'(x)(y-x)\ge-\frac c2(y-x)^2$

Let $f\in C^1(\mathbb R)$ and assume $f'$ is Lipschitz continuous with Lipschitz constant $c\ge 0$. How can we show that $$f(y)-f(x)-f'(x)(y-x)\ge-\frac c2(y-x)^2\tag1$$ for all $x,y\in\mathbb R$? I ...
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Finding an Explicit Formula for a Geometric Series

CONTEXT: Uni question made up by lecturer If you have to take a $100$mg drug every $8$ hours, and just before you take the drug, $20$% of it remains in your body, how would you write an explicit ...
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Taylor series of a function of several functions

How do we find the Taylor series of $f(c,\nabla c,\nabla^{2}c)$ where $c,\nabla c$ and $\nabla^{2}c$ are functions of $x$ in $1$-D. enter image description here
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How to find the radius of convergence of $\sum^{\infty}_{n=1}\frac{n!}{n^n}z^n$?

$$S=\sum^{\infty}_{n=1}\frac{n!}{n^n}z^n$$ Where $z \in \mathbb{C}$. Using D'Alambert's test of convergence: $$\frac{1}{R}=\lim_{n\to \infty}\frac{\frac{(n+1)!}{(n+1)^{n+1}}}{\frac{n!}{n^n}}$$ $$\...
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How to derive Adam Moulton 2 step implicit method using taylor expansion

I have some confusion on the derivation of multistep method using Taylor expansions. For example, we want to derive the linear 2 step Simpson's rule: My professor first write down the scheme of an ...
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2answers
51 views

Taylor expansion of $\prod_{i=1}^n(1-x_i)^{-1/2}$ around 0

What is the Taylor expansion of $$\prod_{i=1}^n\frac{1}{\sqrt{1-x_i}}$$ around $(0,0,...,0)$?I know that we can write it as $$\prod_{i=1}^n\left(1+\sum_{k=1}^{\infty}\frac{(2k-1)!!x_i^k}{(2k)!!}\right)...
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Matrix equations for derivative schemes using Taylors Theorem

In a tutorial, I have been given an expression for the $k^{th}$ derivative (which is a combination of an expression for the $k^{th}$ derivative and Taylors Theorem): $$ \frac{d^{p}}{dx^{p}}f(x) \ = \...
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Taylor polynomials in Fourier Domain?

Which types of functions admit Taylor polynomials in the Fourier domain? Ok, this may sound cryptic. Therefore I will try to explain it more in depth. We want to approximate function $$x\to f(x) \...
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How do we need to apply Taylor's theorem here?

Let $c>0$ $\sigma_n^2:=\frac{c^2}{n-1}$ for $n\in\mathbb N$ with $n>1$ $f\in C^3(\mathbb R)$ with $f>0$ $g:=\ln f$ (and assume $g'$ is Lipschitz continuous), $$s_n(x,y):=\sum_{i=1}^n(g(y_i)-...
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Why does $1/\text{series}$ equal $\text{series with changed signs}$?

Consider the following function and series: $$ f(z)=-\frac{1}{z} \frac{1-\frac{z^2}{2!}+O(z^4)}{1-\frac{z^2}{3!}+O(z^4)} \tag{1}.$$ I've seen on many posts here and in textbooks that the following ...
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1answer
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Domain of Convergence of Taylor Series in Several Variables

I have the following function in 2 variables: $$\frac{1}{1+z_1+2z_2}$$ I have to find it's Domain of Convergence. My initial thought was that it'll simply be $|z_1 + 2z_2|<1$, but then I realised,...
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Taylor Series centered at 2 Question

I need a little bit of assistance with the following problem: Calculate general Taylor Series and the first 3 terms of the Taylor series centered at 2 of the function f(x) = ln(x^2 + 2x + 1) I know ...
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Assume $f$ has continuous derivative on $[0,2]$, and $f(0)=f(2)=1$, if $\lvert f'\rvert \le 1$, prove that $1 \le \int_0^2 f(x)dx\ \le 3$.

I'm working on my textbook's exercise for hours but just can't prove it. This question might has connections with the integral form of the remainder of the Taylor polynomial. By using the first ...
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Taylor's theorem applied to get a randomized approximation of a function

Suppose we have a function $f:\mathbb R\rightarrow \mathbb R$ that is differentiable $n+1$ times at $a$, then Taylor's theorem with Lagrange form for the reminder yields that for any $x\in\mathbb R$, ...
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1answer
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Strange behavior of composite MacLaurin Series

While answering a question about the MacLaurin Series Expansion of a composite function I noticed something strange I can not explain to myself. The task was to verify that the MacLaurin Series ...
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2answers
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Determine whether the Maclaurin series converges absolutely, converges conditionally, or diverges at x = 1.

This is a question from AP Calculus BC practice test. I know this series is the Maclaurin Series for ${ln(x+1)}$ and it would converge on (-1, 1]. I am confused by the phrasing of question b). It ...
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Taylor series of functions with matrix input

Today my professor said something interesting, replace $x$ in $f(x)$ with matrix \begin{bmatrix}x&1\\0&x\end{bmatrix} Then $$f(\begin{bmatrix}x&1\\0&x\end{bmatrix}) = f(x)\begin{...
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Maclaurin series of $\ln(1+x+x^2/2)$ up to order $1$

In the book I'm studying from there's the formula for the Maclaurin series of $\ln(1+x)$: $\ln(1+x) = x - x^2/2 + x^3/3 + ... + (-1)^{n-1} x^n/n + \mathcal{O}(x^{n+1})$ In the examples the book ...
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Determine if $f$ has local maximum or minimum in $0$ by using Maclaurin series

Let $f(x) = \ln(1 + x + x^2/2) - \sin(x)$ Determine if $f$ has a local maximum or minimum in $x=0$. My approach: Set $u(x) = x + x^2/2$ I know the Maclaurin series $\ln(1+x) = x - x^2/2 + x^3/3 + ...
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Limits and Taylor series

Use Taylor Series to show that $\displaystyle \lim_{x \longrightarrow 0} \frac{\ln(1+x)}{x}=1.$ Ok, this is very intuitive, we have $\ln(1+x) = \displaystyle \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n}...
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2answers
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Prove $\pi = 2\sqrt3 \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)3^n}$

We wish to prove the following: Use the Power Series for $\tan^{-1}(x)$ to show that $$\pi = 2\sqrt 3 \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)3^n}$$ We have found that $$\tan^{-1}(x) = \sum_{n=0}^{...
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53 views

why isn't all maclaurin series undefined at 0

every maclaurin series contains $$\frac{f^0(0)\cdot x^0}{0!}$$ well, at $x=0 \rightarrow x^0=0^0=0^1\cdot 0^{-1}=\frac{0}{0}$ which has no arithmetic meaning. I understand it is indeterminate, but we ...
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1answer
38 views

Is it possible to use a Taylor expansion to approximate x2/x1?

I have the expression $\frac{x_2}{x_1}$ which I would like to linearize, ie. ($a.x_2+b.x_1+...$). Can I use a Taylor expansion to do this?
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1answer
50 views

An alternate proof of Liouville's theorem

Suppose that $|f(z)|\leq A+B|z|^M$ and that $f$ is entire. Show that for all coefficients $c_j$ with $M<j$ in its power series extansion are $0$. Attampt: $$ |f(z)|=\left|\sum_{k=0}^\infty c_kz^k\...
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1answer
31 views

If $f$ is entire and bounded by $M$ along $|z|=R$ then $|c_k|\leq{M\over R^k}$ [duplicate]

Let $f$ be an entire function bounded by $M$ along $|z|=R$. Show that the coefficients $c_k$ in its power series expansion about $0$ satisfy $$|c_k|\leq{M\over R^k}.$$ I know that $c_k={f^{(k)}(0)\...