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.

learn more… | top users | synonyms (2)

0
votes
2answers
38 views

Interval of convergence for a power series with $x^{2n}$

By definition, the radius of convergence (which is equivalent to the interval) is: $$R:=\frac{1}{\varlimsup_{n\rightarrow+\infty}\sqrt[n]{|a_n|}}$$ Where $\varlimsup_{n\rightarrow+\infty}$ is the ...
0
votes
0answers
12 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: ...
0
votes
1answer
94 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$ $f_y=x-x^3=0$ ...
0
votes
1answer
57 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: ...
0
votes
0answers
18 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: ...
1
vote
2answers
77 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 ...
1
vote
3answers
40 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 ...
1
vote
1answer
49 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 ...
4
votes
1answer
60 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 ...
0
votes
3answers
83 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. ...
2
votes
1answer
37 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 ...
-1
votes
2answers
46 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 ...
0
votes
0answers
25 views

Reverse Taylor series for sine

I want a little help with reverse Taylor series for sinus if is possible :D .From what I read the formula is: RadOfAngle - RadOfAngle^3*3! + RadOfAngle^5*5! - RadOfAngle^7*7! = Sins value. How can I ...
33
votes
1answer
1k views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
1
vote
3answers
280 views

Two similar method to calculate one equation get different answer

Method1:$$\lim_{x\rightarrow0}({\frac{e^x+xe^x}{e^x-1}}-\frac1x)=\lim_{x\rightarrow0}({\frac{e^x+xe^x}{x}}-\frac1x)=\lim_{x\rightarrow0}(\frac{e^x+xe^x-1}{x})=\lim_{x\rightarrow0}(2e^x+xe^x)=2$$ ...
6
votes
4answers
333 views

How is the Taylor polynomial derived?

I understand how the linear approximation works: $$L(x) = f(x_0) + f'(x_0)(x-x_0)$$ But if we continue this approximation in order to get a more accurate result, how do we get the Taylor polynomial ...
0
votes
2answers
87 views

Question involving Taylor series and continuity

Question: $$f(x)=\lim_{n\rightarrow \infty}\frac{x^{2n}-1}{x^{2n}+1}$$ Where is this function continuous? Trial: I analyzed positive terms of x.For large values of n the function approaches to ...
1
vote
0answers
60 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 ...
1
vote
0answers
61 views

Looking for a way to apply the Taylor Series expansion to find derivatives for a function.

This post references the Riemann-Siegel formula found at here and at here. I am writing a Java program which implements this formula. I am having trouble with the remainder terms. The Riemann-Siegel ...
3
votes
1answer
65 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 ...
1
vote
2answers
77 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 ...
0
votes
2answers
44 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 ...
1
vote
2answers
64 views

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

There was a silly question - how to expand the $\ln{x}$ to Maclaurin series?
0
votes
0answers
30 views

General formula for sinusoidal taylor series centered at any a?

I understand that to find a taylor series centred at a particular a value you need to find a formula for the nth derivative, but this is tricky for cos(x) and sin(x). Is it possible to have a formula ...
0
votes
2answers
78 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 ...
2
votes
5answers
150 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 $f(x)<f(0)$, ...
1
vote
3answers
75 views

How to calculate $\lim \limits_{x \to 0^{+}} (\sin x)^{e^{x}-1} $ with Taylor series?

I want to calculate $\lim \limits_{n \to 0^{+}} (\sin x)^{e^{x}-1} $ by using Taylor's Series, and here is what I did so far, and correct me if I'm wrong: $\sin x = x + o(x)$ $e^{x}-1= x + o(x)$ ...
2
votes
2answers
61 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 ...
0
votes
2answers
1k views

taylor expansion in cylindrical coordinates

If I have a function Y($r$,$\theta$) in cylindrical polar coordinate system, then how do I Taylor expand this function around some point ($r_0$,$\theta_0$)? I want the exact formula for Taylor ...
1
vote
2answers
37 views

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 ...
0
votes
1answer
29 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 ...
1
vote
3answers
93 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 $x<0$, ...
1
vote
1answer
42 views

Taylor Approximation of $\cos(0.02)$

Use a Maclaurin $(a=0)$ polynomial for $\cos{(x)}$ with $3$ nonzero terms to approximate $\cos{(0.02)}$. Also, use the Taylor Remainder Theorem to find a bound on the error $\left(\displaystyle ...
0
votes
2answers
231 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 ...
0
votes
2answers
112 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 $T_5(x)$: ...
1
vote
1answer
51 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 ...
1
vote
1answer
39 views

Bernoulli-like generating function

What are the coefficients of the series for: $$\frac x{e^x+1}$$ It looks similar to the Bernoulli generating function, but the $+$ sign is throwing me off. I already found the series for its ...
0
votes
3answers
415 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 ...
2
votes
0answers
110 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 ...
2
votes
1answer
34 views

Convergence of a sequence of $2\times 2$ real matrices

My Try: So $a_n$ can be written as a series very similar to the taylor series of sin: $\displaystyle a_n=\sum_{k=0}^n \frac{(-1)^k b_k}{(2k+1)!}$ for some $b_k$ to be determined. But it is very ...
4
votes
0answers
100 views

Limit and Taylor Series when non-differentiable

I am stuck on a problem similar to this one. Define $$f(\theta,y)=\frac{g(\theta y)}{\int_0^1 g(\theta x)dx}$$ with $g(0)=0$ and $\lim_{t \to 0}g'(t)=+\infty$. I am interested in $\lim_{\theta \to ...
-1
votes
1answer
32 views

Calculus question - Derivative limit

Using Taylor how can I calculate $(\sin(x^3))^{\frac{1}{3}}$ up until $O(x^{13})$,and one more enquiry: Can the function be differentiated on the real axis and if yes what is it's derivative? $f(x) ...
0
votes
2answers
42 views

advice on using Taylor Series for function approximation

I've recently covered the Taylor Series in my studies and have read through several of the posts here which deal almost exclusively with specific problems and proofs but none seem to be answering a ...
2
votes
0answers
91 views

Validity of approximating a difference equation with a differential equation

Consider the following two equations: $$ \begin{cases} A_k(n+1)-A_k(n)=\beta \displaystyle \frac{ A_{k-1}(n)- A_k(n)}{\alpha+ n} + \delta_{k, \beta} \\ \\ \displaystyle ...
0
votes
1answer
43 views

Trouble with series question from STEP past paper

I have answered all parts of this question but the last part. By using the identity, $\cot x - \tan x = 2\cot 2x$ ...
1
vote
2answers
43 views

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n. from the series expansions of $\sin x$ and $\cos x$, I get that $\tan ...
0
votes
2answers
77 views

Taylor Series Expansion of $\frac{1}{\sin 2x}$ and $\frac{1}{1-e^{-x}}$

How do you find the Taylor expansions of the expressions: $\frac{1}{sin2x}$ and $\frac{1}{1-e^{-x}}$ I'm not sure what to do since all the terms are in the denominator. Thanks in advance for any ...
2
votes
2answers
66 views

Enlarging set where “weighted” MacLaurin series of $\frac{1}{1 - x}$ equals $\frac{1}{1 - x}$

Is it possible to select real values $a_{n, k}$ so that $$f(x) =\lim_{n \to \infty}\sum_{k = 0}^{n - 1} a_{n, k} x^k = \frac{1}{1 - x} $$ for all $x \in \mathbb{R} \setminus \{1\}$ ? Failing ...
3
votes
2answers
331 views

Can the following trick be expanded upon?

Main Question What is the expansion of $d^{1+\epsilon}?$ Background I noticed the following trick (sometimes more laborious) to directly differentiate $ f(x) $ twice without differentiating it even ...
1
vote
1answer
44 views

Expanding a function into a series

I am trying to follow a proof in QFT notes, however I am unable to follow this step - it's basically Laurent/Taylor expansion but I have very little experience with it. It's claimed that: ...