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 Series General Formulas

I'm looking at 2 different Wikipedia pages: The formula here is different than the one given at the end of the section here. Aside from the remainder, why choose one over the other? I'm assuming ...
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0answers
33 views

Taylor expansion for the roots of real polynomials

Consider a (real) polynomial $\mathcal{P}$ in the variable $x$ whose coefficients are themselves polynomials in the parameter $\lambda$. I am searching a taylor expansion in $\lambda$ for the roots ...
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2answers
116 views

What is the justification for taylor series for functions with one or no critical points?

Some(but not all) smooth functions can be represented by taylor series. And the common justification people give why this is possible(like in this question, and that) is something along this line: ...
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3answers
573 views

Under what conditions is integrating over a series expansion valid for an improper integral?

On stackoverflow, a question was asked about getting Mathematica to evaluate the integral, $$\int^\infty_0 \frac{e^{-x}}{\sin x} \, \mathrm{d}x$$ which we know is divergent. In one of the answers, ...
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2answers
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Taylor Series for $\frac{1}{ 1+x+x^2}$

I tried to solve it in a way. The solution did not match. Please tell me where i went wrong. $\cfrac {1} {1+x+x^2} = \cfrac 4 {4+4x+ 4x^2} = \cfrac 4{ 3+(2x+1)^2} = \cfrac 1{\sqrt 3}\cdot\cfrac 4{ 1+ ...
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2answers
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Find Taylor series for $f(x)=e^x$ at $c=3$. Then simplify the series and show how it could have been obtained directly from the series $f$ at $c=0$.

Find the Taylor series for $f(x)=e^x$ about the point $c=3$. Then simplify the series and show how it could have been obtained directly from the series for $f$ about $c=0$. Taylor's Theorem: ...
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1answer
69 views

How can I find the Maclaurin series of $\frac{1}{1+x+x^2}$?

How can I find the Maclaurin series of $\frac{1}{1+x+x^2}$? At first, I found the Maclaurin series of $\frac{1}{1+x}$, which is $\sum_{n=0}^{\infty}(-1)^{n}x^{n}$ and simply replaced $x$ with $x^2 + x ...
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1answer
94 views

How many terms required in $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place?

How many terms are required in the series $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place? Here is what I have: $$e\approx ...
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2answers
97 views

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$.

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$. Taylor's Theorem: $$ f(x)=\sum_{k=0}^n{1\over ...
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0answers
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Original proof of Taylor's theorem

There are numerous proofs for Taylor's theorem, but What's the original proof for Taylor's theorem (by Taylor?)? In Wikipedia it says: Taylor's theorem is named after the mathematician Brook ...
2
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1answer
49 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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2answers
71 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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0answers
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Using Integrals to Derive the Taylor Series

An answerer gave a derivation (Where do the factorials come from in the taylor series?) for the standard form of the taylor polynomial series, copied and pasted below for ease of viewing. I wanted to ...
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0answers
31 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
28 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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1answer
32 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
42 views

General question on Taylor Series

The Taylor Series comes from an assumption that a function has an expression as power series. Given such assumption we can then say that the $n$-th derivative and evaluate them at $x = a$, it can give ...
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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.
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1answer
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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
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Maclaurin series for $\frac{x}{e^x-1}$

Maclaurin series for $$\frac{x}{e^x-1}$$ The answer is $$1-\frac x2 + \frac {x^2}{12} - \frac {x^4}{720} + \cdots$$ How can i get that answer?
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5answers
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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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5answers
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Multivariate Taylor Expansion

I am in confidence with Taylor expansion of function $f\colon R \to R$, but I when my professor started to use higher order derivatives and multivariate Taylor expansion of $f\colon R^n \to R$ and ...
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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 ...
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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: ...
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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 ...
3
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1answer
1k views

Difference in limits because of greatest-integer function

A Problem : \begin{equation}\lim_{x\to 0} \frac{\sin x}{x}\end{equation} results in the solution : $1$ But the same function enclosed in a greatest integer function results in a $0$ ...
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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} ...
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1answer
286 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
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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
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 ...
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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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1answer
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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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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} + ...
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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 ...
2
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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 ...
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: ...
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1answer
76 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
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 ...
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0answers
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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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1answer
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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
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
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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
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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 ...
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1answer
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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. ...
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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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2answers
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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
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 ...