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's theorem with field of rational numbers

Let $\mathbb{Q}$ be the field of rational numbers. Let $\alpha \in \mathbb{R}$ with $1<\alpha<2$ be such that $\alpha^i$ is irrational for every odd integer $i$. Define $f:\mathbb{Q} ...
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1answer
47 views

Finding the Taylor series representation for $\frac{1}{1-z}\quad |z|>1$

I'm tasked with finding the Taylor series representation for $\frac{1}{1-z}\quad |z|>1$ from $\frac{1}{1-z}\quad |z|<1$ where the following hint is provided ...
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1answer
26 views

How large should n be chosen in to have: $|e^x - p_n(x)| < 10^{-15}$?

How large should n be chosen in: to have: $\lvert e^x - p_n(x) \rvert \leq 10^{-15} $, $-1 \leq x \leq 1 $ How can i solve it? A lot of value are missing
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1answer
51 views

Assuming that $y(x)$ can be written as a Taylor series about $x = 0$, find the first three terms in the series solution of the initial value problem

Assuming that $y(x)$ can be written as a Taylor series about $x = 0$, find the first three terms in the series solution of the initial value problem $$y'=y^2+x$$ $y(0) = 1$ I started by letting ...
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0answers
115 views

Need help with finding smallest possible value of a constant $M$ referred to in Taylor's inequality

Let  $f(x) = 1/x,  0.6 ≤ x ≤ 1.4$.  Suppose that we approximate $f(x)$ by the $3$rd degree Taylor polynomial $T_3(x)$ centered at $a = 1$. Taylor's inequality gives an estimate for the error ...
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3answers
64 views

Taylor Series of $f(x) = \sqrt{x}$ about $c = 1$ [duplicate]

Taylor Series of $f(x) = \sqrt{x}$ about $c = 1$ I've tried doing this problem but stuck at finding a pattern.. Work: $$T_n = \sum^\infty_{n=0}\frac{f^n(c)}{n!}(x-c)^n = f(a) + ...
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1answer
56 views

Find the Laurent series for $f(z)=\frac{2}{(z-4)}-\frac{3}{(z+1)}$

The question is 2 parts - I'm to find the Laurent series valid for $$1 < |z| <4$$ and $$|z| > 5$$ I've already solved the first part, but I have a conceptual question about the second part. ...
2
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1answer
24 views

Approximating $\frac{a+\delta a}{b + \delta b}$

I have two quantities $a(t)$ and $b(t)$ that have a constant mean ($a$ and $b$) and some small fluctuating noise part with vanishing mean $\delta a(t)$ and $\delta b(t)$. I'll write them as $a(t) = a ...
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1answer
34 views

Remainder term of Taylor polynomial in matrix calculus

This is my first time asking a question here and it might be rather specific. Hopefully there is still someone who can answer. Assume for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is at ...
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0answers
46 views

Taylor of $\ln(f(exp(x))))$?

Let $ f(x) = \sum a_n x^n$ Such that The $a_n$ are real and $f(a),f ' (a) , f " (a) > 0 $ for any real $a > 0$. Let $ \ln(f(exp(x))) = \sum b_n x^n $. Let $c_n = a_n - b_n$. For a given $f$ ...
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2answers
21 views

Find the Laurent series for $p(4/z)$

Find the Laurent series for $p(4/z)$ given that $p(z)=(z-3)^3$ My attempt: if the Taylor series for $p(z)$ looks like $$\frac{-27}{0!}+ \frac{27z}{1!}-\frac{18z^2}{2!}+ ...
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2answers
44 views

Finding a Taylor's series from f(x) = $x^{3/2}$ at x = 1

My function is $x^{3/2}$ and I need to find the Taylor series around 1. I calculated the first 5 derivatives. And I calculated the values of those results at 1. The derivatives were: ...
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1answer
25 views

Find the laurent series for $e^{2/(z-1)}$

I'm starting to learn about Laurent series. The way I understand it is that it is the same as a Taylor expansion, but with negative terms in addition to the positive terms. I may be wrong, but isn't ...
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1answer
39 views

Is there a word describing the derivatives of an object's motion?

Consider an object moving along a straight line. One might say something about its displacement, velocity or acceleration. These are the 0th, 1st and 2nd derivatives of the object's displacement. ...
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27 views

binomial expansion for non integer exponent $n$

Does there exist an approximation of the term $(x+a)^n$ for non-integer $n$, but for any $x>0$. I've found many topics concerning this but they cite that the binomial expansion is valid only for ...
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1answer
27 views

Matrix Square Root Taylor Series

Does applying the taylor series for the square root to matrices give a meaningful definition for the square root of a matrix? (As in the definition of the matrix exponential?) For symmetric matrices, ...
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3answers
446 views

What function does this Taylor Series represent?

What is the function $f$ who's Taylor series is $1 - \frac{x}{4} + \frac{x^2}{7} - \frac{x^3}{10} + \cdots$ ? I need to find the value of the series $ \sum^{\infty}_{n = 0}a_n = 1 - \frac{1}{4} + ...
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1answer
21 views

Find the power series of the 3 functions together

$$f(z)=3e^z+2ze^z+z^2e^z$$ In the first part of the problem, I had to find the power series of each term, which are listed below. $$3e^z=\sum_{n=0} \frac{3z^n}{n!}$$ $$2ze^z= \sum_{n=0} ...
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0answers
20 views

Sign of approximation error and remainder (residual)

In the Wikipedia article Taylor series it is said that: The error incurred in approximating a function by its $n$th-degree Taylor polynomial is called the remainder or residual and is denoted by ...
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0answers
40 views

Taylor expansion and Ito when the value function has a non differentiable point

I am trying to solve a multi-period free boundary problem, of an Ornstein–Uhlenbeck process, where each stopping decision at each period adds a different constant (penalty or bonus). Solving with a ...
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1answer
17 views

how to find out radius of convergence of the following series?

let $f(x)=\frac{1}{1+x^2}$. Consider its taylor series expansion about a point $a$ in $\mathbb R$ given by $f(x)=\sum_{n=0}^\infty a_n(x-a)^n$. what is the radius of convergence of this series? one ...
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24 views

Does this equation has a closed form solution?

We have $K$ non-negative coefficients: $a_1,a_2,\dots,a_K,A_1,A_2,\dots,A_K$, where $A_i\geq0,\;a_i\in(0,1),\;\sum A_i<T$. The equation is: $$\sum_{i=1}^K\frac{A_i}{1-a_ix}=T,\quad x\in(0,1)$$
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1answer
30 views

Show convergence of remainder term in Taylor's theorem

If $f(x):=e^x$, show that the remainder term in Taylor's Theorem converges to zero as $n \rightarrow \infty$ for each fixed $x_0$ and $x$. My question is, what is the expression for "remainder"? Any ...
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2answers
76 views

Does the Fundamental Theorem of Algebra hold true for infinite polynomials?

I know that by the Fundamental Theorem of Algebra, every polynomial of positive degree has a zero in $\mathbb{C}$. Does this also hold for polynomials of infinite degree? Like, for example, the Taylor ...
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1answer
72 views

Taylor series and extrapolation 2

Hey guys, I've finished the first three parts, but I have no idea how to approach part d and part e. Any hints would help! Thanks! Update: I've finished the first three parts. For part d, I found ...
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0answers
30 views

Performing Taylor series with 'fractional' powers

Taylor expansions are used all the time for physics problems, but sometimes they don't work because we expanded in the 'wrong' parameter. For example, suppose we're doing a relativistic kinematics ...
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1answer
53 views

Perturbation of the Upper Incomplete Gamma Function

The Upper Incomplete Gamma function, for $t \in \mathbb{R}$, is defined as: \begin{equation} \Gamma(α,β)=\int_{β}^{\infty}t^{α-1}e^{-t}dt \end{equation} For the problem which I am studying it takes ...
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1answer
48 views

Why is $\left ( \frac{x}{j}+1 \right )^{-1}\left ( \frac{1}{j}+1 \right )^{x}=1+\frac{x(x-1)}{2j^2}+O(j^{-3})$?

On this page, it says $$ \left ( \frac{x}{j}+1 \right )^{-1}\left ( \frac{1}{j}+1 \right )^{x}=1+\frac{x(x-1)}{2j^2}+O(j^{-3}). $$ Could anyone please enlighten me how it is deduced? I tried to find ...
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2answers
30 views

Simplifying Taylor Series for the function $\sqrt{2+x}$ about $x=3$

I have a past exam with the solution here. But I can't understand how they combined and simplified the terms going from the third last to the second last line. How did they get rid of $[1 \cdot 3 ...
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2answers
63 views

Taylor series expansion of $ f(x)=e^{-x^2}$

How to find Taylor series expansion of $f(x)=e^{-x^2}$ What I'm stuck at is proving that the error $$R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$$ of the expansion tends to zero.
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1answer
50 views

Trapezoid rule error

I am trying to compute the error in the trapezoid rule integration for a function $f(x)$ in the interval $[a,b]$. I believe I have to Taylor-expand $f(x)$ around $x=a$ $f(a) + (x-a)f'(a)+ 1/2 ...
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1answer
33 views

Series expantion of a function around an undefined point

Given a function which is undefined in at least one point, such as $$f(x) = \frac{x^2}{(x-a)(x-b)},$$ how do you find the series expansion about that undefined point? The problem with Taylor ...
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2answers
73 views

Implications of differentiability and Taylor expansion

Consider a function $\phi: \Theta \subseteq \mathbb{R}^l \rightarrow \mathbb{R}$. Fix $\theta_0 \in \Theta$. Assume: (1) $\phi(\cdot)$ differentiable at $\theta_0$ (2) The gradient at $\theta_0$, ...
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1answer
18 views

Taylor expansion of this electric field

I'm trying to determine what happens when R>>z for the below equation $\frac{z\sigma}{2\varepsilon }\left ( \frac{1}{z}-\frac{1}{\sqrt{R^{2}+z^{2}}} \right )$ Like most books, which is a great ...
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0answers
9 views

interpolation with missing gradients

I am just wondering if anyone has encountered a similar problem before. Let's consider a differentiable function in 1 dimension for now. Suppose that this function is expensive to compute. In ...
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2answers
67 views

Series Expansion of $\frac{1}{(x+a)^n}$

Is there a simple series expansion for: $$ \frac{1}{(x+a)^n}$$ Where $|a| < x$ and $x \in \Bbb R^+$ I am interested in a one that does not contain the $(x+a)^n$.
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1answer
48 views

Can you get a closed-form for $\lim_{x\to\infty}\int_2^x-\log\left(1-\frac{1}{t^2}\right)dt$ and evaluate its limit as $x\to\infty$?

I know how deduce for $x>1$ that $$-\log\left(1-\frac{1}{x^2}\right)=\sum_{n=0}^\infty\frac{1}{n+1}\frac{1}{x^{2n+2}},$$ and from a online tool that $$\int -\log\left(1-\frac{1}{x^2}\right)dx ...
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1answer
38 views

What does the range of convergence in Maclaurin series mean?

I tried to calculate the following Maclaurin series: $$ f(x) = \sqrt{1+x^2} = 1+\frac{x}{2}+\sum_{n=2}^{\infty}(-1)^{n+1}\frac{x^{2n}(1)(3)...(2n-3)}{2^n n!} $$ With Ratio Test, I found the range of ...
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2answers
49 views

taylor expansion and limit of a series??

$f(x)=\int_0^xtan^{-1}tdt$ what is the taylor expansion about the origin of this function? and how do i use this to get the limit of the series $1-\frac{1}{2}-\frac {1}{3}+\frac {1}{4}+\frac ...
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1answer
22 views

Generating function with G'(1) and G''(1) reducing to 0/0 = undefined

My question is about an analysis of an algorithm in D. E. Knuth's book The Art of Computer Programming, Vol. 1. More specifically, it is about section 1.2.10, equations 20 to 22. First we have a ...
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2answers
39 views

Third degree polynomial for $ \sin(x^2+y^2)?$

Let R be an open region containing the point $(x_0,y_0).$ Let f, g, and h be functions defined on R, except possibly at$ (x_0,y_0).$ Suppose that for every $(x,y) \in R$ not equal to $(x_0,y_0)$, we ...
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2answers
56 views

Proving divergence of a series via Taylor Expansion:$\sum\left(\sqrt{1+\frac{(-1)^n}{\sqrt{n}}}-1\right)$

I would like to prove using a taylor expansion that the serie $\sum \ \sqrt{1+\frac{(-1)^n}{\sqrt{n}}}-1$ is divergent for $n\geq 1$. What is the expansion to prove it ? Thanks
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1answer
47 views

Show that each biholomorphic function $f:U\setminus M \longrightarrow U\setminus M$ has a biholomorphic extension $g:U \longrightarrow U$.

Let $U\subseteq \mathbb{C}$ be an open and bounded set without isolated points on the border. Let $M\subset U$ be a set without limit points in $U$. Show that each biholomorphic function $f:U\setminus ...
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1answer
18 views

Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
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0answers
46 views

If $c_{n}$ coefficient of the expansion of $f$. Show that $\sum_{n=0}^{\infty}\left|c_{n}\right|^{2}r^{2n}$ is an expression determined by an integral [duplicate]

Let $U\subseteq \mathbb{C}$ be an open set, $z_{0}\in U$ and $R>0$ such that $\mathcal{B}_{R}(z_{0}) \subseteq U$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function with Taylor's serie ...
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0answers
30 views

If $f(z)=\sum_{n=0}^{\infty}c_{n}(z-z_{0})^{n}$, then $c_{n}r^{n}=\frac{1}{2\pi}\int_{0}^{2\pi}f\left(z_{0}+re^{-2nt}\right)dt$

Let $U\subseteq \mathbb{C}$ be an open set, $z_{0}\in U$ and $R>0$ such that $\mathcal{B}_{R}(z_{0}) \subseteq U$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function with Taylor's serie ...
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2answers
47 views

About Taylor series

Suppose $f(0) = 0, f'(0) = 2, f''(0) = −1$ and $|f''' (x)| ≤ 0.024$ for $0 ≤ x ≤ 2$. Estimate $f(1)$ to $4$ significant figures by using a Taylor polynomial. Compute a good bound for the absolute ...
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2answers
40 views

Deducing the series expansion of $\arctan(x^2)$ via the series expansion of $\arctan(x)$ at $x=0$

Comparing the series expansion of $\arctan(x^2)$ and $\arctan(x)$ at $x=0$ it looks like one can take the result from $\arctan(x)$ and replace each $x$ with $x^2$ to deduce the series expansion of ...
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2answers
81 views

Taylor expanding $\frac{e^x}{x}$?

How can you taylor expand $$\frac{e^x}{x}$$ Can it be expanded at $x = 0$? Can it be expanded as $x \to 0$?
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1answer
73 views

Find the limit of this sequence

Suppose $$ f_n(x)=\sum_{k=1}^n \frac{\cos(kx)}{k}, $$ and let $a_n=\min_{x \in [0,\pi/2]} f_n(x)$, find $\lim_{n \to\infty} a_n$. I wrote a program and found that the $\arg\min_{x \in [0,\pi/2]} ...