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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Quality of $E(f(X))\approx f(EX)+\frac 1 2 f''(EX)\sigma_X^2$ approximation

For convex $f$, we have Jensen's lower bound $Ef(X)\ge f(EX)$. What conditions do we need to put on $f,X$ so that the second order expansion in the title would be an upper/lower bound/good ...
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1answer
24 views

How to show $\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}$

How to show the below equation ? $$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)} ~~~~~(t\in \mathbb Z^+)$$
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28 views

Help with taylor series summation?

$f(x) = \sqrt{x}$ When you do expansion, you get the following: $$4 + \frac{x-16}{2^3}\times 1! + \frac{(x-16)^2}{2^8}\times 2! + \frac{3(x-16)^3}{2^{13}}\times 3!$$$$+ ...
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1answer
46 views

Solving Second-order non-linear ODE, with fractional expansions

I am solving a differential equation related to fluid mechanics, a rigid air bubble rising towards the surface of a liquid. Doing all of the maths, I have come to this differential equation, which I ...
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4answers
176 views

Nested radical $\sqrt{x+\sqrt{x^2+\cdots\sqrt{x^n+\cdots}}}$

I am studying the $f(x) = \sqrt{x+\sqrt{x^2+\cdots\sqrt{x^n+\cdots}}}$ for $x \in (0,\infty)$ and I am trying to get closed form formula for this, or at least some useful series/expansion. Any ideas ...
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92 views

Numerical Computation - Taylor series

I'm taking numerical computation course, and I have a problem with this question: Apply Taylor's formula to obtain a power series approximation about $a=0$ to $\sin(\pi x/2)$. Find the remainder, ...
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1answer
295 views

The remainder of Taylor (Maclaurin) series of $\cos(x)$

Something is bothering me with the remainder of the Taylor (Maclaurin) series of $\cos(x)$. The formula of $a_n$ is $(-1)^k \frac{x^{2n}}{(2n)!}$. By Leibniz Theorem, $r_n<a_{n+1}$ which is, ...
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1answer
24 views

Implicit Euler local error issue

I'm not sure I get what's going on here, and online resources are not helpful, at least I didn't find any helpful ones. For the problem: $$ \frac{dy}{dt} = f(t, y(t))$$ a numerical solution for ...
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0answers
24 views

An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
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1answer
21 views

Alternate (approximate) form for Hypergeometric function 1F1(0.5, 1.5, -x)

I have the following Hypergeometric function of the first kind: $_{1}F_1(\frac{1}{2}, \frac{3}{2}, -x)$ where $x$ is not negative. This function can also be written as the following series: ...
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3answers
51 views

Find the laurent series for $\frac{1}{z(z-2)^2}$ centered at z=2 and specify the region in which it converges.

My attempt: $$\frac{1}{z(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{A}{z}+\frac{B}{z-2}+\frac{C}{(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{(1/4)}{z}+\frac{(-1/4)}{z-2}+\frac{(1/2)}{(z-2)^2}$$ This is ...
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1answer
38 views

Taylor Series Expansion of a function?

So, i was studying my Computer Vision lecture notes and i came across this formula which says Say, i have a function $f(x,y,t)$, $x,y$ and $t$ are the varying factors After $t+ \nabla t$, i have ...
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2answers
783 views

Series expansion of a function at infinity

I know it is possible to expand an expandable function for a real, and for infinite by setting $x=\dfrac1y$ and then expanding for $0$. But my question is, how do we do if the evaluation of the new ...
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0answers
23 views

Loglinearize a nonlinear difference equation

There is an exercise in my weekly problem set I have to solve and I am really struggling. The setting is as follows: Suppose you have the following nonlinear difference equation in terms of Π_t and ...
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2answers
28 views

Taylor series of $\ln{\sqrt[4]{\frac{x-2}{5-x}}}$ to $o((x-x_0)^n)$ when $x_0 = 3$

Well I have tried to get it as $$f(x) = f(x_0) + \frac{f'(x_0)(x-x_0)}{1!} + \frac{f''(x_0)(x-x_0)^2}{2!} + ... + o((x-x_0)^n)$$ and got wrong results: First: $$f'(x) = \frac{3}{4(x-2)(5-x)}$$ ...
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1answer
45 views

Computing limits with Taylor series. [duplicate]

I'm posting my whole thought process, but I'd like to ask specifically about whether is my expansion of $\log(1+y)$ into Taylor series good & allowed in this situation? Is there an easier (not ...
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1answer
28 views

Series expansion of $\ln(1+(1-x)^{1/2})$

I am practicing series expansions by coming up with some expression, trying to do it by myself and then checking myself with wolfram alpha. However, I have some issues with the following example I ...
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2answers
36 views

How can I get Maclaurin series for $\frac{x^2 + 3e^x}{e^{2x}}$?

The answer for it is $$3 + \sum_{k=1}^n (3+k(k-1)2^{k-2})\frac{(-1)^k}{k!} x^k + o(x^n)$$ Well, I've tried to change every $e^x$ to $1 + x + \frac{x}{2!} + ... + o(x^n)$ and got nothing useful. I know ...
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1answer
39 views

Finding a neighbourhood interval with Taylor polynomial

I was lectured on this however I did not understand what should I do exactly. How can I find this interval? Find a neighbourhood ($-\delta,\delta$) of $0$ for which the $3rd$ order Taylor polynomial ...
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0answers
31 views

How to calculate higher than second order Taylor series in non-cartesian coordinates?

My question is how to calculate the taylor series of a function in non-cartesian coordinates. For orders of two or less, this is answered in another question (taylor expansion in cylindrical ...
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2answers
24 views

How can I invert the asymptotic form $x^{3/2}=y^{3/2}(1+a/y^2 + … )$ to find $y=y(x)$?

This might sound silly, but the fact there's a $a/y^2$ term in the expansion made me feel a little lost. Could anyone help? Thanks
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1answer
24 views

How to expand $1/(e^{a-b} +1) - 1/(e^{a+b} +1)$ for $a\gg b$? [closed]

I've tried Taylor expansion, but could get nowhere. Could anyone help me out? Thanks
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2answers
27 views

How to expand $(x+y)^{3/2}-(x-y)^{3/2}$ for $x\gg y$

My first and naive impression is that the result is 0 but according to Salinas, Introduction to Statistical Physics that's $3x^{1/2}y + O[(x/y)^3]$ I think Taylor expansion would do it. The thing ...
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1answer
59 views

How to compute $\sum_{n=1}^{+\infty}\frac{1}{n!2^{3n}}$?

I want to find the following sum: $$ \sum_{n=1}^{+\infty}\frac{1}{n!2^{3n}} $$ However I am not sure that the following computation is true: $$ ...
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1answer
29 views

show the Taylor expansion converge in a given condition

By applying the given condition, I obtain the following. But then what I can do to show the error tends to zero.
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7answers
10k views

How are the Taylor Series derived?

I know the Taylor Series are infinite sums that represent some functions like $\sin(x)$. But it has always made me wonder how they were derived? How is something like ...
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1answer
54 views

Taylor Series for even orders

Find the Taylor polynomial of order $2n$ at $0$ of the function $$\ln\left(\frac{1+x}{1-x}\right)$$ Show that the remainder approaches 0 as n increases, if $−1 < x < 1$. Use this to compute ...
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1answer
25 views

Taylor Polynomial for Odd Degree [closed]

Find the Taylor polynomial of degree 2n + 1 at 0 of the function $$f(x)=\int_0^x\frac{sint}{t}dt$$ Any help here would be much appreciated.
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2answers
45 views

In Maclaurin's series why do functions approximate accurately for large $x$?

I understand that Maclaurin's series is about approximate derivatives at the origin, so it makes sense that they approximate well at and near the origin, but why, when you begin to add more terms to ...
3
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2answers
184 views

Approximating $\ln(1+\exp(x)+\exp(y))$

Ok, so I know that $\ln(1+e^x)\approx x$ when $x$ is large. But what about $\ln(1+e^x+e^y)$ when both $x$ and $y$ are large? I can figure out cases when $x\gg y$ or $y\gg x$ since that simplifies ...
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0answers
23 views

Show that Runge Kutta Gauss Legendre is 4th order

So I have this Butcher array of \begin{array}{c|ccc} \frac{1}{2}-\frac{\sqrt{3}}{6} & \frac{1}{4} & \frac{1}{4}-\frac{\sqrt{3}}{6}\\ \frac{1}{2}+\frac{\sqrt{3}}{6} & ...
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2answers
21 views

Lagrange Error of Binomial Approximation

I'm trying to do test corrections for Calculus 2, but I am stuck on this problem. Find an upper bound for your error estimate in part (b) The estimate is of $\sqrt2$ using the 4th degree Taylor ...
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0answers
17 views

Gradient Approximation Methods

I am trying to find a way to approximate the gradient of a multivariate function. This relates to gradient-based optimization problems. My assumptions are as follows: Implicit function (FEA) Very ...
3
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1answer
44 views

Why does $\frac{1}{1-z}=-\sum_{n=0}^{\infty} \frac{1}{z^n}$ if $|z|>1$

Why does $$\frac{1}{1-z}=-\sum_{n=0}^{\infty} \frac{1}{z^n}$$ if $|z|>1$ I know the case for $|z|<1$ and if $|z|>1 \implies |1/z|<1$ so $$\frac{1}{1-\frac{1}{z}}=\sum_{n=0}^{\infty} ...
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2answers
31 views

Technical challenge: Limit of von-Mises distribution approximates normal. How to take the limit?

Background: In psychophysics or the study of ant navigation it's important to represent random variables on a circle. The most popular distribution for doing so is the von-Mises distribution (the ...
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1answer
27 views

Taylor Polynomium and Conic Section

A real function of two variable is given by: $ f(x,y)=exp(x+y)·cos(x-y) $ The approximating polynomium of 2nd degree for f(x,y) with converging point $(x_0,y_0)=(0,0)$ called $P_2(x,y)$ a) Find ...
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0answers
25 views

How do we find a Taylor series for complex numbers?

Suppose I wanted to find a Taylor series for $f(z)=\sin(z)$ centered at $0$ for example. I know for $f(x)=\sin(x)$ we can use the formula $$\sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}$$ ...
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1answer
47 views

Taylor series near $0$ for $(1+x)^{\large\frac{\ln x}x}$

I want to find the Taylor series of $(1+x)^{\frac{\ln x}{x}}$. I have tried to write it down as $e^{\frac{\ln x \ln{1+x}}{x}}$ but it didn't work. Also I tried to use $(1+x)^n =1+nx+n(n-1)x^2 /2$ but ...
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2answers
74 views

quesiton about taylor series

Calculate $f'(0)$, $f''(0)$ and $f'''(0)$ where $$f(x)=(\sin x)·(\cos x)^{1808} ·(\exp x^{601})·(1+3x^3−5x^4 +2754x^{232})$$ I know we should solve it use the Taylor series for sins, cosx, and expo, ...
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1answer
57 views

Question about asympotic expansion of $\int_0^x t\sqrt{ln(t)} dt$

Question: Find the constants $$a_0, a_1, a_2$$ in the asympotic expansion $$\int_0^x t\sqrt{ln(t)} dt$$ = $a_0(x^2)(lnx)^\frac 12$ + $a_1\frac {x^2}{(lnx)^\frac 12}$ + $a_2\frac {x^2}{(lnx)^\frac ...
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2answers
31 views

Taylor series with integration

I am having trouble with this problem. I have used taylor series but dont know what to do next to proof that S = 1;
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1answer
23 views

Expressing the taylor's theorem via integrals for the residual

Let $f:\mathbb{R}^n\to\mathbb{R}$ is twice continuous differentiable and $p\in\mathbb{R}^n$ why do we have that: $ \nabla f(x+p)=\nabla f(x) + \int\limits_0^1 \nabla^2 f(x+tp)p dt$ ?? $\nabla^2f(x)$ ...
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2answers
30 views

Operations with truncation error

Assume I'm given an equation $a\frac{\partial p}{\partial x}+b\frac{\partial q}{\partial x}$. I approximate $\frac{\partial p}{\partial x}$ using a truncated taylor series to first order, such that ...
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4answers
1k views

Trigonometric identities using $\sin x$ and $\cos x$ definition as infinite series

Can someone show the way to proof that $$\cos(x+y) = \cos x\cdot\cos y - \sin x\cdot\sin y$$ and $$\cos^2x+\sin^2 x = 1$$ using the definition of $\sin x$ and $\cos x$ with infinite series. thanks...
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1answer
169 views

Taylor's theorem with field of rational numbers

Let $\mathbb{Q}$ be the field of rational numbers. Let $\alpha \in \mathbb{R}$ with $1<\alpha<2$ be such that $\alpha^i$ is irrational for every odd integer $i$. Define $f:\mathbb{Q} ...
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1answer
51 views

Assuming that $y(x)$ can be written as a Taylor series about $x = 0$, find the first three terms in the series solution of the initial value problem

Assuming that $y(x)$ can be written as a Taylor series about $x = 0$, find the first three terms in the series solution of the initial value problem $$y'=y^2+x$$ $y(0) = 1$ I started by letting ...
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1answer
47 views

Finding the Taylor series representation for $\frac{1}{1-z}\quad |z|>1$

I'm tasked with finding the Taylor series representation for $\frac{1}{1-z}\quad |z|>1$ from $\frac{1}{1-z}\quad |z|<1$ where the following hint is provided ...
0
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1answer
26 views

How large should n be chosen in to have: $|e^x - p_n(x)| < 10^{-15}$?

How large should n be chosen in: to have: $\lvert e^x - p_n(x) \rvert \leq 10^{-15} $, $-1 \leq x \leq 1 $ How can i solve it? A lot of value are missing
2
votes
2answers
73 views

Implications of differentiability and Taylor expansion

Consider a function $\phi: \Theta \subseteq \mathbb{R}^l \rightarrow \mathbb{R}$. Fix $\theta_0 \in \Theta$. Assume: (1) $\phi(\cdot)$ differentiable at $\theta_0$ (2) The gradient at $\theta_0$, ...
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0answers
115 views

Need help with finding smallest possible value of a constant $M$ referred to in Taylor's inequality

Let  $f(x) = 1/x,  0.6 ≤ x ≤ 1.4$.  Suppose that we approximate $f(x)$ by the $3$rd degree Taylor polynomial $T_3(x)$ centered at $a = 1$. Taylor's inequality gives an estimate for the error ...