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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Minimize a convex function over a convex cone

I want to minimize a strictly convex function over a convex cone, where the number of parameter is the same as the sample size. Does the Newton-type algorithm have a global (or local) convergence ...
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1answer
42 views

Taylor series question

I've been struggling with this problem: Find the Taylor series representation for $xe^{2x}$ I was able to find the Taylor series for $e^{2x}$ (centered at a=k) in a previous exercise which I ...
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Normal vector to Taylor polynomial

I have a Multivariable Taylor series approximation of a function which contains up to second order terms. How would I determine the normal vector at the point of evaluation given that I'm using a ...
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17 views

Higher order terms in Taylor expansion tend to infinity faster.

Suppose $g$ is a smooth bounded and symmetric probability density function (pdf). Let $\{(X_1,Y_1), ..., (X_N,Y_N)\}$ be a random sample from the joint pdf $t(x,y)$. Further assume $a\to 0$ and $Na ...
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1answer
19 views

Asymptotic error expansion of global error for single step methods

My question refers to the proof of the following theorem, but it may suffice to just skip the theorem and continue with the problematic taylor expansion $(\ast)$: Let $f(t,y)$ and the single step ...
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2answers
67 views

Is the Taylor series of the $f: \mathbb{R} \to \mathbb{R}$ evaluated at $0$ converges pointwise (in the whole $\mathbb{R}$)?

I would like to solve this problem: Consider the $f: \mathbb{R} \to \mathbb{R}$ function which is $n$-times differentiable for any $n \in \mathbb{N}$. Is it true that the Taylor series of ...
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1answer
30 views

Does $f(x) = ln(1+2x+2x^2) - 2x$ have a critical point at x = 0?

If we taylor expand $f(x)$ we get: $f(x) = \frac{-4}{3}x^3+O(x^4)$ We also know that $f(0) = 0$. The correct answer is no, because f(x) will be negative for positive x close to zero, and positive ...
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2answers
57 views

Find Taylor series of function around $x=0$

I'm trying to calculate the Taylor serie around $x=0$ of the function $$f(x)=\int\limits_0^xe^{-t^2}dt$$ I tried to use the fundamental theorem of calculus, but I'm still stuck.
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1answer
29 views

Multiplicative version of Maclaurin or Talyor series

Is there a multiplicative version of Maclaurin or Talyor series? May be in the format $\ln y = b_0 + b_1 \ln x + b_2 (\ln x)^2 + \cdots $ I want to use that as an approximation in a regression ...
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5answers
130 views

How to find the MacLaurin series of $\frac{1}{1+e^x}$

Mathcad software gives me the answer as: $$ \frac{1}{1+e^x} = \frac{1}{2} -\frac{x}{4} +\frac{x^3}{48} -\frac{x^5}{480} +\cdots$$ I have no idea how it found that and i don't understand. What i did is ...
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Engineering Mathematics Problem with Taylor's Series

This is a problem from Engineering Mathematics book by K.A. Stroud 7th edition, Exercise 18, Chapter 12 Further problems. It has been given in a physics manner, but it just requires manipulation of ...
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1answer
20 views

Taylor expansion of an expectation

Ok guys, I'm reading a book and I'm not getting quite well a concept. If I have to expand $U'(Y_0(1+r_i))$ around $Y_0(1+r_f)$, why I get this: ...
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1answer
21 views

Taylor's series and nth derivitive

The problem is: Calculate the Taylor's series in "$a=1$" of the function : $$f(x)=(5x-4)^{-\frac{7}{3}}\ .$$ I've started off by calculating the $n$th derivative of a function : \begin{align} ...
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2answers
120 views

Bounding $f'$ in terms of $f$ and $f''$

Assume that $f: \mathbb{R} \to [0,\infty)$ is $C^2$ and $|f''(x)| \leq A$ for all $x$. Show that the inequality $$(f'(x))^2 \le 2Af(x)$$ holds for all $x$. The hint given in the question was, ...
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1answer
45 views

Is square root of Taylor series of $f(x)$ equivalent to the Taylor series of square root of $f(x)$

Mathematica treats two expressions as they are equivalent: Sqrt[Series[y[x], {x, x0, 1}]] Series[Sqrt[y[x]], {x, x0, 1}] Is that mathematically justified? Is ...
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1answer
51 views

Finding $f^{(2015)}(0,0,0)(x,y,z)^{2015}$ if $f=xe^{1-xy}+ \frac{z}{1-z^2}+z^{3}\sin(x+y).$

$$f^{(2015)}(0,0,0)(x,y,z)^{2015}$$ $$f=xe^{1-xy}+ \frac{z}{1-z^2}+z^{3}\sin(x+y).$$ I will give you my thoughts as soon as I type out an example from class that makes sense to me. Use of Taylor ...
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1answer
15 views

Understanding central difference formula for computing numerical gradient

More can be found here: http://www.math.ohiou.edu/courses/math3600/lecture27.pdf. I'm having trouble understanding what happens to the $h$ in this example where the central difference error is ...
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1answer
38 views

How to find solutions for this nonlinear equation?

I want to find an analytical solution $x$ as a function of parameters $(e,u,r,t)\in\mathbb{R}^4$ that satisfies the following condition: ...
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1answer
36 views

If all derivatives are zero at a point, what does this imply?

Let's say I have a function $f$ which for all positive $n$ and some complex point $z_0$ satisfies $f^{(n)}(z_0) = 0$. What does this say about the function's analyticity or holomorphicity? Obviously, ...
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2answers
41 views

Calculating $f'(x)$ with $f(x)$ and a relative error?

I want to calculate $f'(x)$ using the formula: $$ f'(x) = \frac{f(x+h) - f(x)}{h}$$. Of course the error here is $o(h)$. However, what if in measuring $f(x)$ and $f(x+h)$ I have a relative error of ...
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0answers
8 views

Expression for variance using Taylor series

I have the following expression for the variance: $$Var[\hat{f_n}(x)]=\frac{1}{2nh}\cdot\frac{(F(x+h)-F(x-h))}{2h}\cdot((1-(F(x+h)-F(x-h)))$$ If $h \downarrow 0$, this is supposed to be equal to: ...
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2answers
33 views

Taylor expansion of $f(x,y)=xy-x+2x^3-yx^3$ about (0,1)…

I am asked to expand $f(x,y)=xy-x+2x^3-yx^3$ about (0,1) up to second order: First I found the required derivatives, and their values at (0,1), $ f_x=y-1+6x^2-3yx^2=0$ $f_y=x-x^3=0$ ...
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1answer
25 views

Taylor Expansion for a two-variable function

I am having a lot of difficulty understanding the given notations for Taylor Expansion for two variables, on a website they gave the expansion up to the second order: ...
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0answers
9 views

Error term Taylor expansion

We have $E[\hat{f_n}(x)]=\frac{F(x+h)-F(x-h)}{2h}$, $h\downarrow0$. In order to compute this expectation I need to use a Taylor expansion, under the assumption that f' and f'' exists: ...
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2answers
71 views

Proving that for any Differentiable distribution $F(x)$, an expression is increasing in $x$?

I am guessing that for a continuous random variable on $[0,1]$, $$ U(x)=\Big[x F(x) + \int_x^1 (1-t)f(t)dt\Big]x $$ is increasing for any distributions, because I can show $$ U'(x)=2xF+x^2f+\int_x^1 ...
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3answers
36 views

calculating the taylor series when there is an integral involved

one of the exercises is to calculate the taylor expansion at x=0 and degree 4 for some function. For example: $$\int_{0}^{x} e^{-t^{2}} dt$$ I actually have no clue how to get started. I know how to ...
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1answer
39 views

Is there any standard method for finding the function defined by a Taylor/Laurent series?

Say you have a Taylor series defined by $$\sum_{n=0}^{\infty}a_nx^n$$ Is there any standard way to figure out what function is defined by the series? One option I see is just looking at the ...
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1answer
47 views

Taylor expansion of $\frac{1}{1-D}$, where $D$ is the differential operator

I understand that we can represent $e^D$ simply as a power series of D. But what about functions of D which are not entire on the complex plane? What if the function has no taylor expansion, or if it ...
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3answers
59 views

Problem with Maclaurin series expansion method.

Look at the following series: 1 + 2x + 3x^2 + 4x^3 + 5x^4 + ..... You can say by using any method that the series is divergent. It indeed diverges but we use this as a series expansion for 1/(1-x)^2. ...
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1answer
23 views

Lévy-Khintchine formula and Taylor expansion

I have trouble finding out why this condition $\int_{\mathbb{R}\backslash\{0\}} \min(1, x^2 ) \nu(dx) < \infty$ in the Lévy-Khintchine formula is necessary. The Lévy-Khintchine formula is ...
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2answers
39 views

How to show $K = O(\frac{\log x}{\log\log x})$ in this case?

How to show $K = O(\frac{\log x}{\log\log x})$ when $K$ is the smallest number for the following inequality to hold: $$ \sum_{k=K+1}^\infty \frac{(\ln2)^{k-1}}{k!} \leq \frac{1}{x} $$ This observation ...
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0answers
41 views

How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
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1answer
39 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
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2answers
43 views

How to expand the $\ln(x)$ to Maclaurin series?

There was a silly question - how to expand the $\ln{x}$ to Maclaurin series?
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2answers
38 views

Logarithms and Taylor Series

Before Log Tables, how were they able to compute expressions such as $2^{2.221}$? I understand they could take a Taylor expansion of $\frac{1}{x}$, but how were they able to condense the expansion ...
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2answers
71 views

How to derive the Taylor expansion form of a polynomial expression?

I want to change this polynomial into the form $\sum_{k=0}^m a_k x^k$ $$q_m(x)=\sum_{k=0}^m(-1)^k\binom{2m+1}{2k+1}x^k(1-x)^{m-k}$$ I see no way to do this as I fear one might need intricate binomial ...
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1answer
205 views

Infinite Product Representation of $\sin x$

I've recently taken interest in infinite products, and I'm having trouble with a proof I found in this PDF file: "Infinite Products and Elementary Functions": An intermediate step in finding an ...
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2answers
37 views

Runge Phenomena and Taylor Expansion

From The Weierstrass Approximation Theorem Vs The Runge's Phenomenon: We contrast this to polynomial interpolation: this is a specific method for generating a sequence of polynomials that ...
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5answers
110 views

how to prove that $\ln(1+x)< x$

I want to prove that: $\ln(x+1)< x$. My idea is to define: $f(x) = \ln(x+1) - x$, so: $f'(x) = \dfrac1{1+x} - 1 = \dfrac{-x}{1+x} < 0, \text{ for }x >0$. Which leads to $f(x)<f(0)$, ...
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2answers
53 views

Asymptotic Expansion of $\ f(x)=(1-\beta \frac{ log(log(x))}{log(x)})^{\beta}$

So I got this function and I'm looking for an asymptotic expansion for different values of$\ \beta > 1 $ $\ f(x)=\left(1- \beta \frac{\log \left( \log(x) \right)}{\log(x)} \right)^{\beta}$ as $\ x ...
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2answers
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computing maclaurin series for $(\sin x)^3$ , order $3$

I have a clarification to ask: I want to compute $f(x)=(\sin x)^3$ by maclaurin series, order $n=3$. I know that: $\sin x=x-\dfrac{x^3}{3!}+R_3(x)$. So can i say that: $\sin^3x=(\sin ...
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1answer
21 views

Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...
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3answers
92 views

proving that $g(x)=0$ has one real root

Given $g(x)=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^{2n+1}}{(2n+1)!}$, Need to prove that $g(x)=0$ has one real root. I thought to use the fact that $e^x<T_{2n}(x)$ for all $x<0$, ...
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0answers
24 views

Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and approximate $R_n < \frac{1}{10000}$

I am tasked with the following: Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and use the power series to approximate $\displaystyle \int_{0}^{0.5}3e^{-x^2/2}$ with error ...
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2answers
25 views

Find MacLaurin polynomial of integral

I have not the slightest idea how to begin with the following problem. My first thought is to integrate it before trying to find the MacLaurin polynomial, but I don't know if that is possible. Here is ...
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2answers
26 views

Help understanding a question

I know this probably isn't the best question to post as far as further use with others, but I literally have no where else to turn to for study assistance. My problem is as follows: Find $T_5(x)$: ...
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1answer
24 views

Approximating an integral with taylor series

I am working on the following homework problem: "Assume that $\sin(x)$ equals its Maclaurin series for all $x$. Use the Maclaurin series for $\sin(5x^2)$ to evaluate the integral ...
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0answers
44 views

Taylor expansion in terms of known Taylor series

Let $f(x)$ be a "nice" function with the following properties: $f$ is a real analytic, strictly increasing, odd and bounded function, i.e. $f(-x)=-f(x)$ and $-1<f(x)<1$. Further, let $f(x)>0, ...
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3answers
25 views

Finding an expression for the general term of a taylor series

I am working on a homework problem that asks the following: "Find an expression for the general term of each of the series below. Use $n$ as your index, and pick your general term so that the sum ...
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0answers
90 views

About sparse polynomial squares

Given $p\in\mathbb{Q}[x]$, we define the weigth of $p$ as: $$ W(p) = \#\{n\in\mathbb{N}: [x^n]\,p(x)\neq 0\} $$ i.e. as the number of non-zero terms. By playing a bit with the Taylor series of ...