# Tagged Questions

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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### Why can't I get good approximation when choosing values away from point of expansion? (Taylor series)

I was in the middle of doing a computing project assigned to me when I came across the question. $\operatorname{P}_N(x)$ is the taylor polynomial for $f(x)=\ln(x)$ expanded around pouint $x_0=1$ ...
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### How to compute $\lim\limits_{x\to 0}\dfrac{e^{f(x)}-e^x}{2x-\sin\left( f(2x) \right)}$

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a function such that : $f(x)=x-x^3+o(x^3).$ Compute $$\lim\limits_{x\to 0}\dfrac{e^{f(x)}-e^x}{2x-\sin\left( f(2x) \right)}$$ My thoughts: ...
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### Behaviour of $f(x)=e-\left(1+\frac{1}{x} \right)^{x}$ when $x\to+\infty$

This is from an MCQ contest. For all $x\geq 1$ let $f(x)=e-\left(1+\dfrac{1}{x} \right)^{x}$ then we have : $f(x)\mathrel{\underset{_+\infty}{\sim}}\dfrac{e}{x}$ and $f$ is integrable on ...
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### Integrability of $f(x)=\left(1+\frac{1}{x} \right)^{1+\frac{1}{x}}-a-\frac{b}{x}$

This is from an MCQ contest. For all $x\geq 1$ let $$f(x)=\left(1+\dfrac{1}{x} \right)^{1+\dfrac{1}{x}}-a-\dfrac{b}{x}$$ note that ...
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### Does Cauchy's estimate imply analyticity?

Komatsu says here (Proc. Japan Acad. Volume 36, Number 3 (1960), 90-93) that a smooth function which satisfies Cauchy's estimate is analytic. How does one prove this? Surely, if Cauchy's estimates ...
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### Taylor expansion for $\arcsin^2{x}$

I stumbled upon this particular expansion that was included in this post. $$\displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$ This caught ...
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### Finding closed-form approximations of the solutions of $f(x,y)=0$

Consider $$f(x,y)=\sum_{i=1}^n\dfrac{\sin(\omega_ix)}{\sin(\omega_iy)}r_i$$ where $n,\omega_i,r_i>0$ are known parameters. Restrict to domains where $\sin(\omega_iy)\neq 0$, and by symmetry ...
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### When is $a(z) = b(c(z))$?

Let $a(z)$ be a given transcendental entire function. When is $a(z)=b(c(z))$ where $b,c$ are also transcendental entire functions ? How to find such $b,c$ ? In particular when $a$ is given by a ...
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### Central Difference taylor approximation

We are asked to show that We have so far managed to show the first two equalities using finite difference approximations but the last one still eludes us. Any hints?
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### Generalised Taylor series to fractional order derivatives and special functions

A year ago or so I read this papar which was wonderfully illuminating link. For example the author seduces the reader with wonderfully compact representations like that of the bessel $J_v(z)$ function ...
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### Looking for Taylor series expansion of $\ln(x)$

We know that the expansion of $$\sin(x)$$ is $$x/1!-x^3/3!\cdots$$ Without using Wolfram alpha, please help me find the expansion of $\ln(x)$. I have my way of doing it, but am checking myself with ...
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### Coefficients of series

Suppose that i have a function $f(x)=\sum_{i=0}^{\infty}a_ix^i$ with radius of convergence $r_f>0$ and that i want to write $f$ in a form $f(x)={e^{g(x)}}$ where $e$ is natural logarithm base and ...
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### Taylor expansion $\frac{y-(y+1) \log (y+1)}{y^2 (y+1)}$

I would like to use Taylor expansion around $y=0$ for that expression but i got stuck $$\frac{y-(y+1) \log (y+1)}{y^2 (y+1)}$$ Taylor expansion around $y=0$ ...
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### Maclaurin expansion of $y=\frac{1+x+x^2}{1-x+x^2}$ to $x^4$

Maclaurin expansion of $$\displaystyle y=\frac{1+x+x^2}{1-x+x^2}\,\,\text{to } x^4$$ I have tried by using Maclaurin expansion of $\frac1{1-x}=1+x+x^2+\cdots +x^n+o(x^n)$, but it seems not lead ...
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### Techniques in analytic number theory

I'm fairly new to the subject and trying to figure things out. Would be nice to hear some ideas and trickery for what follows. Suppose we wish to show there exists an integer $x$ in some finite set ...
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### expansion of Bessel function $J_1$

this is a general question. Is there a general way to expand the Bessel Function $J_1(z)$ when $z\in \mathbb{C}$ and when z is large? Or in other words, what is the asymptotic expansion of $J_1(z)$? ...
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### If $y'(x) =y(x)-x +1$ and $y(0)=1$ then $y^{(n)}(0) = 1$ for every $n\ge2$

Start of the taylor series: f(x) = f(0) Which from the question is 1. f(x) = 1 f'(0) = y - 0 + 1 The current y value ...
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I solved the problem by using a known series: $\frac{1}{\sqrt{1-x^2}}$, but the solution I got is wrong. Also, I'm not sure what to do with the constant of integration $C$. Where is my mistake? $$... 1answer 26 views ### Taylor series expansion and radius of convergence The problem is: Expand the given function using Taylor's expansion around a=1. f(x)=(5x-4)^{-7/3}, and then find the radius of convergence of the obtained series. Hint : Write the nth derivative ... 1answer 190 views ### Estimate probability of event using moments of a distribution or a Taylor expansion involving the moments Let's say we have four moments (\mu_1, \mu_2, \mu_3, \mu_4) of a probabilty distribution of a random variable X and the goal is to get the probability \rm{P}(X \leq t) for a certain value of ... 1answer 22 views ### Limit Question involving logarithmic taylor expansion I need to evaluate the limit for part of my proof:$$ \lim_{n \to \infty}\left(1-\dfrac{1}{\eta^{x}} \right)^n My attempt: \begin{align*} \lim_{n \to \infty} F_{\eta_n}(x) &= \lim_{n \to ... 3answers 93 views ### How to show \sum_0^\infty \frac{x\lambda^x} {x!} = \lambda e^\lambda? I know that \sum_0^\infty \frac{\lambda^x} {x!} = e^\lambda, but I'm having a really difficult time dealing with the extra x. 2answers 158 views ### Taylor Series Expansion/Small Angle Approximation I'm working on my first physics problem set (mainly math review) and I'm having a really hard time with one of the questions: Trigonometry: Start with the general expression: f(x)=a+bx+cx^2+ ... 1answer 75 views ### Expression for the gradient using Taylor's Theorem I've just started reading Nocedal and Wright's book on Numerical Optimization. On page 14 there is a formula for the value of the gradient in some point (equation 2.5) that I cannot derive myself. ... 2answers 44 views ### Taylor expansion of a random variable I'm struggling a little with this expansion: Where E is the expectation operator, U is a function of Y and Z^~ is a random variable. In the second passage why the expansion looks like ... 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 ... 0answers 30 views ### Deriving Taylor theorem expression In one book, I've got a following written: Substituting for f′(x) in (4.15), we obtain the second approximation:f(a +h) \approx f(a) + \int_a^{a+h}[f'(a) + (x-a)f''(a)]dxf(a ...
Consider a remainder of some Taylor series: \begin{align}\frac{Mx^6}{C} + \frac{M'x^8}{C'} + \frac{M'' x^{10}}{C''} + ...\end{align} I want to replace this with $\mathcal{O}(x^\alpha)$ for the best ...
Let $f(x)=(1-x)^{-1}$ and $x_0=0$. Find the nth taylor polynomial $P_n(x)$ for $f(x)$ about $x_0$ Find a value of n needed for $P_n$ to to approximate $f(x)$ to within $10^{-6}$ on the interval ...