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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2
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2answers
59 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.
0
votes
1answer
31 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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4answers
106 views

What does it mean that a taylor series generated for a function f(x) doesnt converge to f(x)?

If a some function f(x) is continous and has derivatives of all orders on some interval I, and assuming that f(x) can be expressed as a power series on I. And now you generate a taylor series for ...
3
votes
5answers
150 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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2answers
105 views

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 ...
1
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1answer
27 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: ...
0
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1answer
28 views

Taylor Series substitution giving different answers

I was given the function: $f(x) = 1/(1+x)^2$ and its Taylor series: $1 - 2x + 3x^2 - 4x^3 + \cdots$ In order to get the Taylor series for the closely related function $1/((1/2)+x)^2$, I simply ...
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1answer
27 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} ...
7
votes
2answers
139 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, ...
1
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1answer
70 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 ...
6
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1answer
213 views

How to find all roots of the quintic using the Bring radical

Finding one root $x_1$ of the quintic equation $x^5 + x = -a$ by using the Bring radical is described on Wikipedia. The root is $x_1 = -a +a^5 -5a^9+35a^{13}+ \ldots$ , and it is found by reversion ...
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1answer
66 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 ...
1
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1answer
33 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 ...
0
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2answers
91 views

Finite difference differentiation formula

I'm trying to understand how the co-efficients of finite differences are calculated. In particular I'm interested in the first derivative for a uniform grid of unit width. I found this document ...
2
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1answer
46 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: ...
1
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1answer
75 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, ...
2
votes
2answers
42 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 ...
0
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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
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1answer
49 views

Trapezodial Rule Error Proof (taylor)

I search for a proof of the (local) error of trapezodial rule using taylor series. I can only find proofs for the error of the rectangle rule and for trapezodial it's always just "similar" whatever ...
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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
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1answer
90 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
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1answer
56 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
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: ...
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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 ...
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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 ...
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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
59 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
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
3answers
78 views

Taylor Series of $\sin x/(1-x)$

Ιs there any fast way to calculate the first four non-zero terms a Taylor Series $\dfrac {\sin x}{1-x}$ at $x=0$ without making big derivatives calculations? I know that $$\sin x = x- \frac{x^3}{6} + ...
2
votes
1answer
32 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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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 ...
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2answers
45 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 ...
1
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3answers
271 views

Two similar method to calculate one equation get different answer

Method1:$$\lim_{x\rightarrow0}({\frac{e^x+xe^x}{e^x-1}}-\frac1x)=\lim_{x\rightarrow0}({\frac{e^x+xe^x}{x}}-\frac1x)=\lim_{x\rightarrow0}(\frac{e^x+xe^x-1}{x})=\lim_{x\rightarrow0}(2e^x+xe^x)=2$$ ...
0
votes
2answers
84 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 ...
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0answers
58 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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0answers
59 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
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1answer
60 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
62 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
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 ...
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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 ...
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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 ...
6
votes
1answer
285 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
69 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
139 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 $f(x)<f(0)$, ...
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 ...
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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
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1answer
27 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
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
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1answer
41 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
150 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 ...