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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using Taylor series to prove an inequality

Prove that if $p^T▽f(x_k)<0$, then $f(x_k+εp)<f(x_k)$ for $ε>0$ sufficiently small. I think we can expand $f(x_k+εp)$ in a Taylor series about the point $x_k$ and look at $f(x_k+εp)-f(x_k)$, ...
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1answer
64 views

Taylor s inequality

Apply Taylor´s inequality to derive the quadratic Taylor approximation of $e^x$ at $x=0$. Could anyone help me out? I tried looking up the definition but I am not sure what is meant by "at $x$ is ...
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1answer
146 views

Bounding approximation error for Taylor polynomial

I've got this problem: Let $f(x) = e^x$. If we aproximate $f(x)$ by $P_4(x)$ in $x_0 = 0$ at $(-r, r)$, find $r \gt 0$ so that the error in the approximation is $\lt 10^{-5}$ What I did is: 1) ...
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1answer
92 views

Exact expansion of functions

Prove that for any twice differentiable function $f: {R}^n \to R$, $f(y) = f(x) + \nabla f(x)^T (y-x)+ \frac{1}{2} (y-x)^T \nabla^2f(z)(y-x) $, for some $z$ on the line segment $[x, y]$. Note that ...
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1answer
186 views

The second order approximation of the Taylor expansion of Characteristic functions:

Let $X$ be a random variable with continuous density $\rho(x)$. Assume that $X$ is symmetric and $\vert X\vert<L$. Since it has a bounded support, all moments of $X$ are well-defined. Let $m_i$ ...
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1answer
323 views

Taylor / Maclaurin series expansion origin. [closed]

Soo we all know Taylor series expansion formula for expansion around expansion point $A(a,f(a))$: $$f(x) \approx \underbrace{f(a)}_{1st~term} + \underbrace{\frac{f'(a)\, (x-a)}{1!}}_{2nd~term}+ ...
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1answer
543 views

Finding the 4th order Taylor expansion of $g(t)= t^3 + 2t^2 + 2t + 1$

Given the function $$g(t) = t^3 + 2t^2 + 2t + 1$$ I would like to find the 4th order expansion of $g(t)$ at $t=t_1$. So far, I have performed the differentiation of $g$, up to $g'''(t)$ w.r.t. $t$, ...
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3answers
896 views

Where do the factorials come from in the taylor series?

Unfortunately, I don't have much detail to give here. But is the general idea to cancel out the constant obtained from taking the derivative. For instance, say my function was ...
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1answer
61 views

Expansion of $x^{-1/2}$ at $0$

Regard the function $f(x) = x^{-1/2}$ on the non-negative real line. The point $z=0$ is 'distinguished' because it is the boundary of the domain of $f$, and because $f$ has a pole there. It seems ...
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5answers
123 views

Proof that $2.82<\pi<3.19$

Using taylor expansion of $\cos$ function. What I have is $$1-\frac{x^2}{2}+\cdots-\frac{x^{4n-2}}{(4n-2)!}<\cos(x)<1-\frac{x^2}{2}+\cdots+\frac{x^{4n}}{(4n)!}$$ How would I proceed from here? ...
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2answers
84 views

Find $\displaystyle\lim_{x \rightarrow 0} \frac{e^{\sin x} - e^x}{\sin^3 2x}$

I have to find $\displaystyle\lim_{x \rightarrow 0} \frac{e^{\sin x} - e^x}{\sin^3 2x}$ using Taylor polynomials. Here's what I've done so far: $e^x = 1 + x + \frac{1}{2} x^2 + \frac{1}{6} x^3 + ...
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2answers
290 views

Calculate the improper integral and the taylor series of $f(x) = \int_{x}^1 \frac{tx}{\sqrt{t^2-x^2}} \,dt$

For the given function $$f(x) = \int_{x}^1 \frac{tx}{\sqrt{t^2-x^2}} \,dt$$ with -1 < x < 1. Calculate the improper integral. Calculate the Taylor series of $f(x)$ at $x=0$ until the third ...
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3answers
1k views

How to expand $\tan x$ in Taylor order to $o(x^6)$

I try to expand $\tan x$ in Taylor order to $o(x^6)$, but searching of all 6 derivative in zero (ex. $\tan'(0), \tan''(0)$ and e.t.c.) is very difficult and slow method. Is there another way to ...
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2answers
517 views

Taylor expansion for $\sqrt{x+2}$

I'm enrolled in Coursera's calculus with a single variable and am trying to solve one of the homework problems. In lecture, it was stated that to expand $\sqrt x$ about $x=a$, you would have: ...
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1answer
161 views

Complex Taylor series and Bernoulli numbers

Let: $f(z)=\frac{z}{e^z-1}$ if $z\ne0$, and $f(z)=1$ if $z=0$. Please help to prove that $\sum_{k=0}^{n-1}\binom n kf^{(k)}(0)=0$ for any $n>1$ and $f^{(2n+1)}(0)=0$ for any natrual $n$.
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1answer
143 views

Taylor's Theorem Application Question, $f(x)$ smooth and $f(0)=0$ implies $f(x)/x$ smooth.

I am wondering the following fact, and I believe I know the answer, but I am not sure why. If $f(x)$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}$, if $f(0)=0$, is it true that $f(x)/x$ is ...
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1answer
70 views

What's incorrect with this Taylor series derivation?

Let's \begin{align} f(T)= &f(0)+ \int_0^T f' (t)dt\\ f(T)=&(0)+f' (T)T-\int_0^T f'' (t)tdt\\ f(T)=&(0)+f' (T)t-f'' (T) \frac{T^2}{2}+\int_0^Tf''' (t) \frac{t^2}{2} dt\\ f(T)=&f(0)+f' ...
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1answer
277 views

When finding upper bound for error, can $\xi$ be different from $x$?

The question is to find $P_3(x)$ for $ f(x) = (x-1) \; \ln x $ about $x_0 = 1$ and find the upper bound on the error for $P_3(0.5)$ used to estimate $f(0.5)$. I got $$ f^{(4)}(x)= \frac{2}{x^3} + ...
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5answers
122 views

$\lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $ solution?

I recently took an math exam where I had this limit to solve $$ \lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $$ and I tought I did it right, since I proceeded like this: 1st I ...
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2answers
571 views

Using Taylor series expansion as a bound

I have a function $f(x)$ that has convergent Taylor series expansion around $x=0$ in the following form: ...
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2answers
66 views

show $\ln \frac{1-x}{1+x}$ is $L^2$ but not $L^1$ using its taylor expansion

I am trying to show that $$ \ln{\Big|\frac{1-x}{1+x}\Big|} $$ belongs to $L^2({\Bbb{R}})$ but not to $L^1({\Bbb{R}})$ by using it's taylor expansion (this is the entire statement of the problem). ...
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2answers
99 views

Behavior of differential equation as argument goes to zero

I'm trying to solve a coupled set of ODEs, but before attempting the full numerical solution, I would like to get an idea of what the solution looks like around the origin. The equation at hand is: ...
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1answer
71 views

Taylor theorem for a multivariate BV function

Given a function $f(x_1,x_2,\ldots,x_n):\Omega\to\mathbb{R}$ which is in $BV(\Omega)$ and has, in $\textbf{0}$, all partial derivatives up to order $n-1$, all equal to 0: $$ \frac{\partial^{|\alpha|} ...
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3answers
395 views

Taylor Series for functions $f:R^n\rightarrow R^n$

I've been told that we can write a taylor series for functions $f:R^n\rightarrow R$ but we can't write one for $f:R^n\rightarrow R^n$. I'm not quite sure why this not possible, but I suspect it have ...
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1answer
229 views

taylor's formula in $\mathbb{R}^n$ as exponential of derivative operator

I hope this question is not to vague. There is some relationship between the formal operator $$e^\frac{d}{dx} = 1 + \frac{d}{dx} + \frac{1}{2!}\frac{d^2}{dx^2} + \ldots $$ and taylor's formula. Is ...
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1answer
268 views

Integration on the unit sphere

I have an integral on the unit sphere as follows. $$I(\mathbf{s}_1, \mathbf{s}_2) = \int_{\mathbb{S}^2} f(\mathbf{x} \cdot \mathbf{s}_1)f(\mathbf{x}\cdot\mathbf{s}_2)d\mathbf{x} $$ where the ...
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3answers
128 views

Taylor expansion of $\sin(x-x^2)$

I am solving a problem that involves expanding $\sin(x-x^2)$. Since $$\sin(x)=x-x^3/6+x^5/120-...$$ i try to substitute $x$ to $x-x^2$ to arrive at expansion of $x-x^2-x^3/6+x^4/2-x^5/2+x^6/6+...$ ...
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2answers
115 views

$f(y) \leq f(x)+\nabla f(x)\cdot (y-x) $ and $f(x)\geq 0$ implies that $f$ is constant.

Here is the question. Suppose that $f: \mathbb R^n \rightarrow \mathbb R$ has two derivatives and the associated hessian matrix is negative semidefinite on all of $\mathbb R^n$. Show that for any ...
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1answer
81 views

Integration using Taylor approximations

I am stuck on two problems: 1) Prove that $$\int_0^1 \frac{1+x^{30}}{1+x^{60}}dx=1+\frac{c}{31}$$ where $0< c <1$. 2) Prove that ...
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2answers
171 views

The Taylor series of $\int_0^x \operatorname{sinc}(t) dt$

I tried to find what is the Taylor series of the function $$\int_0^x \frac{\sin(t)}{t}dt .$$ Any suggestions?
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3answers
167 views

Inverting $\frac{\log(p-1)}{\log\ p}$

I've been searching for an approximation that will allow me to solve for $p$ in $\frac{\log(p-1)}{\log p}$. Since it is so "obviously" $1 - \frac1{(\text{something})}$ for larger $p$ I thought it ...
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1answer
303 views

Is the Taylor series comparable to Fourier series and spherical harmonics?

I am currently trying to grasp spherical harmonics and try to digest that we proved that the sine and cosine functions are a basis for the $L^2$ space of the squared-integrable functions. So as far ...
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6answers
203 views

How to simplify this expression by division

How to divide it $${\frac {{x}^{n-2}-{y}^{n-2}}{x-y}}$$ to remove the $x-y$ term from the denominator. We may assume that $n>2$ is an integer. Thanks.
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2answers
62 views

Prove this equation

I'm taking a course on stochastic analysis. I'm stuck on the very first problem of the lecture notes: $\lim_{n \to \infty} \left(1+\frac{\lambda}{n} + o(n^{-1})\right)^n = \exp(\lambda)$ Prior to ...
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2answers
87 views

showing $f$ is entire

Show that the function $f(z)$ is define by $f(0)=1$ and $f(z)=z^{-1}\sin z $ when $z\neq0$, is entire. $\sin z=z-z^3/3!+z^5/5!-z^7/7!...$ (*) We can write $\sin z=z+z^2g_2(z)$ which $g_2(z)$ is ...
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1answer
118 views

Laurent series of $$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1}=? $$

How to find Laurent series of g(z) ? $$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1} \hspace{10mm} \begin{cases} n \in N \\ 0<a<1 \end{cases} $$ answer is : $$ ...
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1answer
191 views

How to expand $x \sqrt{4 - x}$ to Maclaurin series?

Here is the task: using standard expansions, expand $f(x) = x \sqrt{4-x}$ to Maclaurin's series. I calculated derivatives up to $f^{(5)}(x)$, and got some results. Fortunately, in Maclaurin's ...
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3answers
75 views

A Taylor Expansion Problem

Show that for all $x>0$ we have $\ln(1+x)>x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}$. I know it has to do with taylor expansion but somehow I cannot prove it rigorously.
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1answer
722 views

Lagrange Error Given a Fourth Derivative

I just would like to check my work with someone else's: The function f has derivatives of all orders for all real numbers, and the fourth derivative of f equals e^(sin(x)). If the third-degree Taylor ...
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2answers
95 views

counting the number of real zeros, and find limitation

Let$$P_n(x)=\sum_{k=0}^{n}(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$ let $c_n$ be the number of real zeros of $P_n$. determine$$\lim_{n \rightarrow \infty}\frac{c_n}{2n+1}$$
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Is it possible to algebraically prove that the $n$th degree Taylor remainder of $f(x)$ is less than $K|\Delta x|^{n+1}$ for $K \in \mathbb{R^+}$?

I found a purely algebraic proof, given below, that for a mononomial $f(x) = x^n$ the magnitude of the error of its linear approximation $| f(x) - [f(a) + f'(a)(x-a)] |$ is less than $K(x-a)^2$ for ...
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1answer
185 views

Orthogonal complete set of functions

Every square-integrable function on an interval can be written as a linear combination of e^inx (Fourier series). Are there any other orthogonal and complete set of functions for square integrable ...
3
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1answer
55 views

Exercise about MacLaurin's polynomial and small-o

In class the professor wrote the following limit: $\lim_{x\to 0} \frac{\sinh^2 (x) -x^2}{x^4}$ So he "expanded" (sorry for my English) the MacLaurin's formula for $\sinh x$ up to the 3rd power, and ...
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3answers
967 views

Taylor expansion of $\arccos(1-x)$ around $x=0$ to two terms

Doing a normal Taylor expansion of $\arccos(1-x)$ around $x=0$ to two terms by taking derivatives doesn't work because of division by zero. I've put this into wolfram alpha: ...
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1answer
121 views

Differentiating power series

Consider the power series $$\sum_{n=0}^\infty{\frac{x^{2n}}{(2n)!}}$$ From this, it follows that its sum defines an infinitely differentiable function $f$, given by ...
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1answer
62 views

Combining error terms from two Taylor expansions

When deriving the five-point differentiation formula as shown in this book, the IVT was used to combine $ f^{(5)} (\xi_1) $ and $ f^{(5)} (\xi_2) $ into one error term, $ f^{(5)}(\tilde{\xi}) $ As ...
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2answers
570 views

Taylor Polynomial of $\arctan$ of given Degree and Error

Replace the following function by its taylor polynomial of the given grade, and approximate the error in the given interval: $$f(x) = \arctan(x) \textrm{ by } T_3(f,x,0) \textrm{ in } |x| ...
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1answer
89 views

Taylor polynomial of sin of given degree and error

Replace the following function by its taylor polynomial of the given grade, and approximate the error in the given interval: $$f(x) = \sin(x) \textrm{ by } T_3(f,x,0) \textrm{ in } |x| ...
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2answers
878 views

Find the Taylor Series for $f(x)$ centered at a given value $a$

$$f(x) = \frac{6}{x}\,\, \mathrm{at}\,\, a = -4 .$$ Assume that $f$ has a power series expansion. Do not show that $R_n(x) -> 0$ I took the derivatives of f(x): $$f(x) = 6/x$$ $$f'(x) = -6/x^2$$ ...
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2answers
139 views

How to show $\prod_{i=1}^n (1-p_i) = \exp{(-\sum_{i=1}^n p_i)} + O(\sum_{i=1}^n p_i^2)$

I have a (seemingly) simple question. How can I see (rigorously) that \begin{equation} \prod_{i=1}^n (1-p_i) = \exp{\left(-\sum_{i=1}^n p_i\right)} + O\left(\sum_{i=1}^n p_i^2\right) \end{equation} ...