Questions tagged [taylor-expansion]

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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13 views

Time and space discretization of one dimensional advection equation

I have the following one-dimensional advection equation: $\frac{\partial u}{\partial t}+c\cdot\frac{\partial u}{\partial x}=0$ and I would like to discretise this equation after time and space (...
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1answer
37 views

Derivative of a taylor polynomial with respect to the center

Consider a taylor polynomial of degree $n$ and centered at $a$ :$$T_n(x,a) = \sum\limits_{k=0}^n\dfrac{f^{(k)}(a)}{k!}(x-a)^k$$ Differentiating $T_n$ with respect to $a$ gives a nice result : $$\frac{...
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Show that for $ n \in N $, if a function $f(x)$ is $o(x^{n+1})$, then $f(x)$ is $o(x^n)$

By $o(x^n)$, I mean the little-o of $x^n$, not to be confused with the big-O notation. The definition of $o(x^n)$ in my book is: In general, for any natural number n, if a function $f(x)$ satisfies $...
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Taylor’s Theorem confusion

I can follow the proof but what I do not get is that $g(t)=0$ If we replace $f(t)$ in $26$ by $f(t)$ defined $25$. So $g(t)=0$ for all $t$. Why does that make sense?
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Use Taylor Polynomial to approximate a definite integral.

i have this problem: Get the Taylor Polynomial of grade 3 of $\frac{1}{1-x}$ at the point a=0 Then use the polynomial to approximate the definite integral: $$\int_0^\frac{1}{10} \frac{1}{1-x^2}$$ ...
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Taylor expansion on Spherically Symmetric Manifold

A manifold $M$ is said to be spherically symmetric if, in terms of geodesic polar coordinates $(r,\theta)\in (0,\infty)\times \mathbb{S}^{n-1}\equiv M\setminus \{o\}$ the Riemannian metric $ds^2$ of $...
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Taylor series approximation for $(x^{2}+(1-x)^{2})^{y} - (1 - x)^{2y}$

It is said $1-(1-x)^y \approx yx$ Assuming $yx<<1$ and using Taylor series approximation. $x$ can be $10^{-15}$ and $y$ can be $10^{9}$ so $yx<<1$ My question is can we simplify can we ...
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1answer
28 views

Mary L Boas Ch1, Section 15, Problem 29

$\frac{F}{W} = \frac{T \, \sin \theta}{T \, \cos \theta} = \tan \theta = \theta \; + \; \frac{\theta^3}{3} \; + \; \frac{2\theta^5}{15} \; + ...$; How to solve part b? Solution in a book is $ \frac{x}...
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Taylor series at infinity

I'm required to make a Taylor series expansion of a function $f(x) = \arctan(x)$ at $x = +\infty$. In order to do this I introduce new variable $z = \frac{1}{x}$, so that $x \to +\infty$ is the same ...
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Help making sense of Taylor series using MVT only

I know the coefficients in Taylor series are cooked up to be equal to the derivatives of the original function. But it's still a bit vague to me how information at one single point give the entire ...
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Taylor series for $\frac{1}{\sqrt{(1-v^2)}}$

I don't seem to get the answer the book "The Geometry of Spacetime" by Callahan does e.g. $1 + 1/2(v^2) + O(v^4)$ on Pg. $100$ and it is rather crucial for the ensuing discussion
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Evaluate: $ \lim_{x\to 0}\frac{(1+x)^{1/x}-e+\dfrac{1}{2}ex}{x^2}$ [duplicate]

Evaluate: $$\displaystyle \lim_{x\to 0}\dfrac{(1+x)^{1/x}-e+\dfrac{1}{2}ex}{x^2}$$ My attempt: This limit can be written as $$\displaystyle \lim_{x\to 0}\dfrac{e^{\frac{1}{x}\ln (1+x)}-e+\dfrac{1}{...
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1answer
24 views

Equivalence of different taylor expansion of the same identity

I am trying to compute a second order Taylor expansion of the following identity: $x = yz$. Where I get stucked is when I transformed the original expression into $y= \frac {x}{z}$, I simply get a ...
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42 views

Maclaurin expansion for $~x^a~(1-x)^b~$ [on hold]

How to use a Maclaurin Series Expansion on $~x^a~(1-x)^b~$? There is a singularity at $~x = 0~$ when derivatives are taken. Thank you so much!
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1answer
57 views

Laurent Series of $~\tan(z)~$ expanded in $\frac{\pi}{2} < |z| < \frac{3\pi}{2}~$?

As we know, we can get Laurent series of $~\tan(z)~$ expanded in $~0 \le |z| \lt \frac{\pi}{2}~$ by dividing Taylor series expansion of $~\sin(z)~$ by Taylor series expansion of $~\cos(z)~$, and we'...
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4answers
30 views

Taylor-expansion through change of variables

I need to make a Taylor series expansion about $x = 2$ of the function: $$f(x)=\frac{1}{1-x^2}$$ The general formula for Taylor series is: $$f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\...
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2answers
70 views

Why this approximation not complete with me?

consider the sequence $$ M \sim \sqrt{2\pi} (\frac{n}{e})^n \sqrt[6]{8n^3+4n^2+n+\frac{1}{30}-\frac{1}{K_1n+K_2+\frac{T_1}{n}+\frac{T_2}{n^2}+\frac{T_3}{n^3}}} $$ prove that $$ M \sim \sqrt{\pi} (\...
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2nd order approximation for nondifferentiable convex function

Consider $f$ to be a convex multivariate function from $\mathbb{R}^N$ to $\mathbb{R}$, not differentiable (but, by definition, left- and right-differentiable). Does some kind of "directional" 2nd-...
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Third term in expansion of multivariate cumulant generating function

I'm doing a Taylor series expansion around $\mathbf{0}$ on $$\log{\text{E}[2\mathbf{x}^\intercal\Sigma_{xy}\mathbf{Y}]}$$ where the expectation is over random vector $\mathbf{Y}$. The derivatives ...
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Show multivariable limit does(not) exist possibly using Taylor expansion

I'm supposed to show whether the following limit exists or doesn't exist and if it exist calculate its value. $$\lim_{(x,y,z) \to (0,0,0)} \frac{\ln(1+x^2+y^2+z^2)}{\sin(x^2+y^2+z^2)+xyz}$$ My ...
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Partial sum square root of reciprocal of primes

I would like to know if the following reasoning makes sense. I want to bound/estimate the following sum $$ \sum_{p\leq x}\frac{1}{\sqrt{p}} $$ Using integration by parts we have \begin{align} \sum_{p\...
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Generator Brownian Motion

I'm referring to the construnction of the generator of the Brownian Motion. I marked the points I'm referring in the following here. Why do we need to consider $C_0^2$? Where did we show that $B$ is ...
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proof of second derivative test for function of n variables

Assume that $f : \mathbb{R^n}\to\mathbb{R}$ is a $C^2$ function and that $\mathbf{a}$ is a point such that $∇f(\mathbf{a}) = 0$. Assume also that $H(\mathbf{a})$ has at least one negative eigenvalue ...
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55 views

Sums of infinite algebraic numbers that are $\mathbb{Q}$-linearly independent

While working with infinite sums, I thought of the following problem Consider the sequence $\{a_n\}_{n=1}^{\infty}$ of algebraic numbers that are $\mathbb{Q}$-linearly independent. Is it possible ...
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45 views

Why cant I square the series when doing maclaurin expansion

So the task says: Do the Maclaurin expansion of the given term: $$f(x) = \frac{x}{(1-x)^2}$$ what I did was: https://ibb.co/19Ddd8d The way i tried to solve the task was wrong, I know that i should ...
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Asymptotic expansion of $t$-statistic and $f$-statistic?

I want to mathematically understand the effect of skewness and kurtosis in regards to $t$-statistic and $f$-statistic, for test of means and variances. I read from here, Under regularity ...
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511 views

“Distance” between summands in Taylor Series of Cosine

The Taylor series of cosine is $$\cos(x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}=\frac{x^0}{0!}-\frac{x^2}{2!}+\frac{x^4}{4!}\mp\ ...$$ If we now plot the summands (ignoring the sign and the ...
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1answer
62 views

Can we have a series representation of order $5$ at $x=0$ of the inverse function of $xe^{x^{2}}$?

Well, I am a freshman, This was my homework. With a bit of googling, I found that: $$f^{-1}(x) = \pm \frac{\sqrt{W(2x^2)}}{\sqrt{2}}$$ What our professor did was: $$ f(x)=xe^{x^{2}}=x+x^3+\dfrac{x^5}...
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What’s the derivative of these equations in Taylor series?

I am struggling understand the linear approximation and Taylor.series. Could you give me a hint what are the derivatives of these functions? $$a_2(x_1-x_0)^2 + a_3(x_1-x_0)^3?$$ If it’s stated that $...
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1answer
58 views

Expansion of prime zeta function near singularities

The prime zeta function has the following expansion near its singularity at 1: $$P(1+\varepsilon) = -\ln \varepsilon + C + O(\varepsilon)$$ It also has a singularity at the reciprocal of every ...
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What is the type of a function given by a Taylor series.

Given a general function $f:\mathbb{R}\rightarrow \mathbb{R}$ we might say it has type $\mathbb{R}^\mathbb{R}$. But if it can be written as a Taylor series: $f(x)=\sum\limits_{n=0}^\infty a_n x^n$ ...
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Prove that a function is zero

Suppose $f \in C^{\infty}[-1, 1]$, $f^{(k)}(0) = 0, k = 0, 1, 2, \dots$ and there is some $C \in \mathbb{R}$ such that $\sup_{x \in [-1, 1]}|f^{(k)}(x)| \leq k!C$ for any $k \in \mathbb{N}$. Then $f$ ...
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1answer
48 views

How to obtain the function knowing its higher derivatives at $0$

does some one knows how to obtain $f(x)$ knowing that in x=0 they have the following value $f^{n}(0)= \frac{1}{n-s}$ if $ n=1,3,5,\cdots$ and $f^{n}(0)=0$ otherwise
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1answer
20 views

The Puiseux series and gamma function

Just a thought: In the Taylor expansion of an analytic function $f(x)$, the $\Gamma(n+1) = n!$ appears in the coefficient for $x^n$. So if we use a Puiseux series instead, would we get a $\Gamma(n/k)$ ...
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Error bound for Taylor's Expansion

Use Taylor's Theorem to approximate an upper bound for the error approximation. Sinh$(.9) \approx .9 + \frac{(.9)^3}{3!}$. From the formula, $R_4(x) \leq \frac{|.9|^5}{120}$|cosh$(.9)$|. However, ...
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Are we allowed to apply a (measure-theoretic version of) Taylor's theorem here?

Let $f$ denote the density of $\mathcal N(0,1)$ $g:=\ln f$ $r_n(x):=\frac1{n-1}\sum_{i=2}^n{g'(x_i)}^2$ and $s_n(x):=-\frac1{n-1}\sum_{i=2}^ng''(x_i)$ for $x\in\mathbb R^n$ and $n\ge2$ $c:=\int{f'(x)}...
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Using Taylor(Maclaurin) expansions at infinity

To the best of my knowledge, we usually say that the expansion is most accurate around the expansion point (say, around $x=0$ in the case of the Maclaurin series). I know the question is most likely ...
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Maclaurin series of $\tan(x+x^2)$ to order 3

I know that the maclaurin series of $\tan(x)$ is $\tan(x)=x+\frac{1}{3}x^3+\frac{2}{15}x^5+...$, then shouldn't be $\tan(x+x^2)=(x+x^2)+\frac{1}{3}(x+x^2)^3+\frac{2}{15}(x+x^2)^5+...$? Mathematica ...
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Exponential equation (volatility matching in binomial tree model of option pricing).

Hull's book $\textit{'Options, Futures and Other Derivatives'}$ says that if we solve this equation $e^{r \Delta t}(u+d)-ud-e^{2r\Delta t} = \sigma^2 \Delta t$ with respect to $u$ and $d$, we get $...
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2answers
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Proof of an exponential inequality

This seems to be obvious but I am having a hard time proving it. Any insight greatly appreciated. Statement: Prove for every $b,x \in \mathbb{R}$ such that $b\geq 1$, $|x|\leq b$, it holds that $$(...
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What is the Taylor Expansion of a function [closed]

What is the Taylor expansion of the function $$G(x)=\frac{1}{(1-x^{50})(1-x^{25})(1-x^{10})(1-x^5)(1-x)}\quad ?$$
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1answer
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What is the radius of convergence for the Taylor expansion of $\frac{e^x\sin(x)}{x^2+25}$ about $x=0$?

I would like to know how I could go about finding the radius of convergence for the Maclaurin series of $\frac{e^x\sin(x)}{x^2+25}$. I am familiar with how to find the radius of convergence of more ...
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1answer
33 views

Compute $\lim_{(x,y) \to (0,0)} \frac{x^2 - 2\cos(y) + 2}{y^2 - 2\cos(x) + 2}$

Does the limit $$\lim_{(x,y) \to (0,0)} \left( \frac{x^2 - 2\cos(y) + 2}{y^2 - 2\cos(x) + 2} \right) $$ exist? I think it does and it's equal to $1$, but I don't know how to prove it. I tried to use ...
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How do I approximate $\sum_{i=2}^b \frac{\csc(i x)}{i}$?

If I take the power series around point $b$ of the following sum: $\sum_{i=2}^b \frac{\csc(i x)}{i}$ Series[Sum[Csc[i *x]/i, {i, 2, b}], {x, b, 3}] (...
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10 views

Taylor expansion of small parameter

If I have an expression $$\cos^2{(\bar\theta(1+\sigma m(x_3/l))} $$ where $$\bar\theta <<1$$ and $$m(x_3/l)$$ is a N(0,1) random variable, can I expand in small theta to obtain $$\cos^2{(\bar\...
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21 views

Order of Taylor polynomial in numerical derivative

I'm stuck on a question regarding numerical methods. The aim of this question is to approximate $ f''(x) + f'(x) $ with the highest possible order of truncation error using $Q(h) = \frac{\alpha_{-1} ...
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2answers
44 views

Expanding random N(0,1) variable

If I have an expression $$\frac{1}{1+\sigma m(z/l)}$$ where $m(z/l)$ is a random $N(0,1)$ variable, $\sigma$ is dimensionless, can I rewrite this via an expansion to bring up the random variable on ...
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1answer
47 views

Proving limit of f(x) - Tnf(x) (Taylor) is zero, in multivariable calculus

So as we all probably know, for $ f : \mathbb{R} \to \mathbb{R}$, and $T_nf(x)$ the taylor polynomial at a given point a, we have $\lim\limits_{x \to a} \frac{f(x) - T_nf(x)}{(x-a)^n} = 0$. Our ...
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4answers
48 views

Stuck trying to find the value of this limit using Taylor series.

So I'm having trouble trying to find the value of the limit $$\lim_{x \to 0}\frac{(\sqrt{2}-\sqrt{1+\cos(x)})(e^x-e^{-x})}{\sin^3(x)}$$ using Taylor series (the problem explicits it). Well, this is ...
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1answer
30 views

Taylor series and calculating a sum with its help

So this function is given: $$f(x)=xe^{-2x}$$ I have to calculate its Taylor series around $x=1$ and use it in some way to calculate the following sum: $$\sum_{n=1}^{\infty} \frac{n+2}{(2n)!!}$$ I ...