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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$(1+1/n)^n (1+x/n)$ decreasing iff $x\geq\frac{1}{2}$

$(1+1/n)^n (1+x/n)$ decreasing iff $x\geq\frac{1}{2}$. My question is how to prove $(1+1/n)^n (1+x/n)$ decreasing, starting from $n=1$. For large $n$, it is easy to show by Taylor expansion.
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Why the expected value of error Taylor series approximation around the mean is zero?

I came across the following sentence in Paul Wilmott introduces quantitative finance. ... a random variable S (in our example the stock price), then ... using a Taylor series approximation around the ...
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If Complex Numbers Describe a Circle and Split-complex Numbers Describe a Hyperbola, Can One Make a Hypercomplex Number System to Describe any Shape?

I was thinking about other complex-like systems the other day, and I decided to define a number $o$ such that $o^2 = 1, o \ne \pm 1$. I wondered if there was a formula like Euler's formula for this ...
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How to prove cosine law using the power series expansion of cosine and sine?

How to prove the equation $\cos(x + y) = \cos(x) \cos(y) -\sin(x) \sin(y)$ using the power series expansions \begin{equation*} \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}, \qquad \sin(x)...
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Find the Taylor series at $a=\frac{\pi}{2}$ for $f(x)=x\sin(x)$

I'm trying to find the taylor series at $a = \frac{\pi}{2}$ for $f(x)=x\sin(x)$ The problem is that I don't know if my answer is right.. Could somebody check/correct this $f(x) = x\sin(x)$ let $x = t+\...
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Finding radius of convergence of Taylor Series of $\frac{1}{1+x^3}$about $x = \frac 12$

Let the Taylor Series of $f(x)=\frac{1}{1+x^3}$ about $x=\frac 12$ be $\sum_{n=0}^{\infty}a_n(x-\frac{1}{2})^n$, where $a_n=\frac{f^{(n)}(\frac{1}{2})}{n!}$. Then to find the radius of convergence of ...
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A strange trigonometric inequality

I am trying to check that: $$f(t) = t^2(t-3) + 2 e^{-t/2} + e^{t/2}\left(\sqrt{3} \sin \frac{\sqrt{3}t}{2} + \cos \frac{\sqrt{3} t}{2} \right) - e^{-t/2} \left(\sqrt{3} \sin \frac{\sqrt{3}t}{2} + 3\...
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How do I obtain this expression for the Taylor series of$\frac{z^2}{\sin ^2(z)}$?

Reading about Complex Analysis, I came across the following: Consider first the representation $\frac{\pi ^2}{\sin ^2(\pi z)}=\sum_{n\in \mathbb{Z}}\frac{1} {(z+n)^2}$, which applies for all $z\in \...
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Maclaurin series, find the tenth derivative

The problem is as follows: Find the Maclaurin series of $$\begin{cases} \frac{\sin(x)}{x},& x \neq 0 \\ 1,& x=0 \end{cases}$$ and then find $f^{10}(0)$. I figured out the series, if $x\neq 0$ ...
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Proof check inequality $|$ln$((1+x_1)(1+x_2))-x_1-x_2|\leq 2||x||^2$

Given that $||x|| \leq \frac{1}{2}$, show that for all $x\in \mathbb{R}^2$ the following estimation holds: $|$ln$((1+x_1)(1+x_2))-x_1-x_2|\leq 2||x||^2$ What I did: $|$ln$((1+x_1)(1+x_2))-x_1-x_2|$ $=|...
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How to express a function into powers of $(x-1)$ and $(y-2)$ using Taylor's formula?

Use Taylor's formula to express the following in powers of $(x-1)$ and $(y-2)$: $f(x,y)=x^3 + y^3 + xy^2$ Solution: $f(1,2)=1 +8 + 4=13$ $f_x (1,2) = 3 + 4=7$ $f_y (1,2) = 12 + 4=16$ $f_{xx} (1,2) = 6$...
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Find Taylor series for $\cosh z \cos z$

Find Taylor series for $\cosh z \cos z$. $\cos z = \cosh iz$ and $\cosh z \cosh iz = \dfrac{1}{2}(\cosh (i+1)z + \cosh (i-1)z)$ and finally $$\cosh z \cos z = \dfrac{1}{2}\sum_{n=0}^\infty {\dfrac{(i+...
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Understanding Real analyticity

I'm going to state my assumption of the definition of Real analyticity, and how I understand it based on my current understanding. Please tell me if they are correct or not and please help me ...
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shifting polynomials in Fast Multipole Method

There is one thing I don't get about the FMM algorithm (of coulombic potential in 2D - https://cims.nyu.edu/~donev/Teaching/WrittenOral/Projects/JasonKaye-WrittenAndOral.pdf). Suppose we have ...
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Intuition behind Lagrange remainder term in Taylor's Theorem

Let $f:\mathbb R\to\mathbb R$ be an $n+1$-times differentiable function. Taylor's Theorem states that for each $x\in\mathbb R$, $$ f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\dots+\frac{f^{(n)}(0)}{n!}x^n+\...
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Expand [a+b*log(1-x)]^{-1} as power seires in x

Consider the function $[a+b\ln(1-x)]^{-1}$ near $x=0$, where $a$ and $b$ are constants. As well-known, the Taylor series of $\ln(1-x)$ is $\ln(1-x)=-x-\frac{1}{2}x^2-\frac{1}{3}x^3-\frac{1}{4}x^4+...$....
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Continuity of Taylor remainder for a multivariate $C^1$ function

Suppose I have a function jointly $C^1$ in two variables. By Taylor's theorem I can expand in one of the variables, say the second, and get a remainder term $R(x,y,h) = \frac{f(x,y+h) - f(x,y)}{h} - \...
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Finite differences and Finite Element Method

I have the following 1D problem: \begin{equation} \begin{cases} -u''=f \ \ \ x\in(0,1)\\ u(0)=u(1)=0 \end{cases} \end{equation} I have derived the Galerkin formulation and I have implemented a code ...
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Can someone explain what the 2nd order taylor polynomial of this function is with the remainder [closed]

This is the function: f(x,y) = xcos(πy) - ysin(πx) and the polynomial should be around (1,1)
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Is there a closed-form solution by using talor series approximation? [closed]

I wanna find a closed-form solution for $\mu$ from this expression in terms of other variables: $$BK_L\alpha(K_S+(1-\mu)I_0)^{\alpha-1}=\alpha(K_L+\mu I_0)^{\alpha-1}+s $$ Could Taylor approximation ...
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How to linearize a state space equation with higher order $>2$?

Let us consider the following nonlinear polynomial system $$\dot{x} = f(x,u),$$ where $x=[x_1, ... , x_n]$. A Taylor expansion about $(x_0,u_0)$ gives $$f(x,u) = f(x_0,u_0) + \frac{\partial f}{\...
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How do you find this limit with a relationship to $e$ using Taylor series?

The limit in question is $$ \lim_{x \to 0}\left(\frac{\sin(x)}{x}\right)^{1/x^2} $$ When I replace $\sin(x)$ with its Taylor series about $0$ and cancel out the $x$, I get $$ \lim_{x \to 0}\left(1-\...
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Derriving the taylor series for $e^{\frac{\ln{(x+1)}}{x}}$

So I'm trying to find the Taylor polynomial of $e^{\frac{\ln{(x+1)}}{x}}$ around $x=0$ up to order $1$, however I encounter quite the problem. We know that: $$\ln{(1+x)}=x-\frac{x^2}{2}+O(x^3)$$ Hence,...
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Calculate the expression from an infinitely differentiable function

Calculate the expression from an infinitely differentiable function: enter image description here
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How to evaluate the sum of $\sum_{n=0}^{\infty}\frac{1}{3n^{2}+4n+1}$

I hava an infinite sum $$\sum_{n=0}^{\infty}\frac{1}{3n^{2}+4n+1}$$ I factored the denominator $$\sum_{n=0}^{\infty}\frac{1}{\left(3n+1\right)\left(n+1\right)}$$ Then I separated the fraction $$\frac{...
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How do I find the partial sum of the Maclaurin series for $e^x$?

In one of the problems I am trying to solve, it basically narrowed down to finding the sum $$\sum^{n=c}_{n=0}\frac{x^n}{n!}$$ which is the partial sum of the Maclaurin series for $e^x$. Wolfram | ...
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Is it possible for a function $f(x+a)$ to have a regular Taylor series expansion centered around $a$ instead of $-a$?

I have a question, but I think there's a typo in part (b). If we've got function $f(x+a)$, shouldn't the series end up being centered around $-a$, and not $a$? I've worked it out myself and I'm ...
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Identifying $a_n$ in a Taylor series

Let $f(x) = \sin(x)$ and $g(x) = \cos(x^3)$. Let $h(x) = \sin(x) + \cos(x^3)$. The function $h$ can be represented as the power series: $$h(x)=\sum_{n=0}^\infty a_nx^n$$ which of course is its Taylor ...
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The Maclaurin series of $1-(1-\frac{x^2}{2} + \frac{x^4}{24})^{2/3}$ has all coefficients positive

It was shown in a previous post that the Maclaurin series of $1 - \cos^{2/3} x$ has positive coefficients. There @Dr. Wolfgang Hintze: has noticed that the truncation $1- \frac{x^2}{2} + \frac{x^4}{24}...
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Some positive Maclaurin series arising from estimates of $\exp(-x)$

The Taylor series at $x=0$ for the function $\exp(-x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^n}{n!}$ has alternating sign terms. Consider the "upper" estimates $S_{2k}(x) = \sum_{n=0}^{2k} (-1)...
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Taylor expansion for the Euler's formula

I would like some help to show this: for any $x \in \mathbb{R}$ and $n \in \mathbb{N}$ $$e^{ix} = \sum_{k=0}^{n-1} \frac{(ix)^k}{k!} + \theta \frac{|x|^n}{n!}$$ with some $\theta \in \mathbb{C}$, and $...
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Find particular solution for $(D^2+1)y=e^{a \cos x}$, where a is an arbitary constant.

I tried solving the problem as in the image by taking partial integrals and also by series expansion, but it is becoming more complex to continue and find a close form of it. Please solve it.
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general term of taylor series

I have a taylor series (in this case maclaurin as its at 0) for $(x+1)^\frac23$ and have found the first 5 terms, however, I'm unsure how to find the general term. $f(x) = (x+1)^\frac23$ $\;f'(x) = \...
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Maclaurin series $f\left(x\right)=\frac{1}{\sqrt{1+x^2}}$ [closed]

Construct a Maclaurin series for the function and also give the convergence domain, the general term and 4 worked out terms: $f\left(x\right)=\frac{1}{\sqrt{1+x^2}}$
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Laurent series of this function

Find the Laurent series of the function$$ f(z) = \frac{z+1}{z(z-4)} $$ in the annulus $0<|z-4|<4$. My approach: $$ \begin{aligned} f(z) &= \frac{z+1}{z(z-4)} =\left(\frac{-1}{4}\frac{1}{z}+\...
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Figuring out a proof that $\mathbb{P}\{-\pi \leq X \leq -\pi + \delta\}=\frac{\delta}{2\pi}+\frac{\delta^{k+1}}{(k+1)!}f^{k}(-\pi+) + o(\delta^{k+1})$

The article I am currently reading is Intrinsic Means on the Circle: Uniqueness, Locus and Asymptotics by Hotz and Huckerman, pp. 4-5. Suppose that $X$ is a random variable living in the unit circle ...
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1 answer
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Need advice on approximating the sum of a trigonomitric series which (I think) has no analytical solution

I've encountered a maths problem in a programming project I'm working on. I've tried a lot of things already, and I'm feeling very swamped with maths that is way above my head. I need to find: $$f_N(x)...
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Maclaurin of $(x^2+x+1)^{-1}$

Good day! I stumbled upon this question on Maclaurin series which puzzled me quite a bit. Let $f(x)=\frac{1}{x^2+x+1}$ and that it can be represented by $$\sum_{n=0}^{\infty}{c_nx^n}.$$ Find the value ...
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Some positive Maclaurin series related to $\cos$ and $\sin$ functions

Consider the known inequality $$1 - \cos x \ge 0$$ for all $x\in \mathbb{R}$. This is because the expression equals $2 \sin^2 \frac{x}{2}\ge 0$. Using the power series expression for $\cos x$, the ...
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Complex Logarithm, Taylor Bound

working my way through Shiryaev's book on probability theory, I have encountered an identity for the principal value of the logarithm: for any complex number $z$ with $|z|<\frac12$, we have $\log(z+...
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Pointwise upper bound on $|f(x+y+z)-[f(y)+f(z)+f'(y)(x+z)]|$ where $f(x) = |x|^{p-1}x$

Let $f(x) = |x|^{p-1}x$ for some $2 \leq p \leq 3$. I've seen the pointwise estimate $$\left| f(x+y+z)-[f(y)+f(z)+f'(y)x + f'(y)z] \right| \lesssim |f(x)| + |f'(x)y| + |f'(z)x| + |f'(z)y|$$ in Lemma 3....
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Maclaurin series of $\frac{x^2}{1- x \cot x}$

I wonder if there is an explicit formula for the Maclaurin expansion of $\frac{x^2}{1 - x \cot x}$. We know an explicit formula for $1- x \cot x$. Due to the continued fraction formula for $\tan x$, ...
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1 vote
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Analytic tanh-like function with infinite radius of convergence.

Consider the Taylor expansion of $\tanh$ around $0$. The radius of convergence is finite ($\pi/2$). Define a $\tanh$-like function a function $f:\mathbb R\to\mathbb R$ such that: $f(0) = 0$; $\lim_{x\...
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Doubt in multivariate Taylor's theorem

The way wikipedia presents it, I don't understand how many functions $h_\alpha$ do we have in the remainder term? Using matrix notation, $$f({\boldsymbol {x}})=f({\boldsymbol {a}})+\nabla f(\...
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Second order Taylor expansion of $\int_0^\mu\frac{x}{2\pi} - F(x)dx$ when $F(x)$ is a distribution function

I'm currently reading an article in which the authors perform a second order Taylor series approximation $\int_0^\mu\frac{x}{2\pi} - F(x)dx = \frac{\mu^2}{4\pi} - \frac{\mu^2}{2}F'(0) + o(\mu^2)$ when ...
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Find $f^{(n)}(3)$ from the equation $\frac{(-1)^n(2)^nn!}{(3+2x)^{n+1}}$ [closed]

Given $\displaystyle f(x) = \frac{1}{3+2x}$ and $a = 3$, find $f'(x)$, $f''(x)$, $f'''(x)$, $f''''(x)$, then for general $n$. a. Find the formula for $f^{(n)}(x)$ and simplify. b. Find $f^{(n)}(3)$ ...
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Computing limits using Taylor expansions and $o$ notation on both sides of a fraction

Let's define $o(g(x))$ as usually: $$ \forall x \ne a.g(x) \ne 0 \\ f(x) = o(g(x)) \space \text{when} \space x \to a \implies \lim_{x \to a} \frac{f(x)}{g(x)}=0 $$ In theorem $7.8$, Tom Apostol in his ...
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$h(x) = o(f(x)g(x)) \implies h(x) = f(x) \cdot o(g(x))$?

Let's define $o(g(x))$ as usually: $$ \forall x \ne a.g(x) \ne 0 \\ f(x) = o(g(x)) \space \text{when} \space x \to a \implies \lim_{x \to a} \frac{f(x)}{g(x)}=0 $$ Is it true that: $h(x) = o(f(x)g(x)) ...
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For what infinite series can a closed form be obtained by means of the $\text{Sum} = \text{Product} $ method?

Euler solved the Basel problem by equating the Taylor series and the infinite product representation of $\sin(x)/x$: $$\sum_{n=1}^{\infty}(-1)^{n}\frac{x^{2n}}{(2n+1)!} = \prod_{k=1}^{\infty}\bigg{(} ...
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Partial derivatives in scalar field taylor expansion

$\newcommand{\v}[1]{\mathbf{#1}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\dd}[1]{\mathrm{d}#1}$ In our lecture notes we derived the following formula for the Taylor expansion of a scalar ...
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