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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Derivation of multivariable Taylor series

I am having trouble grokking why it is, assuming that the function is analytic everywhere (and many other assumptions that I am, no doubt, naively assuming), that this is true: ...
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42 views

Taylor Series and equation

I have this equation: $$960 - 84.60 \cdot \frac{1-(1+i)^{-12}}{i} = 0$$ I simplify $( 1+i)^{-12}$ with a Taylor series $( 1 + x)^a$. but I obtain $i = 0.087201167$ but the real result should be $i ...
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1answer
53 views

Taylor expansion of $((H+\epsilon A)^T R^{-1} (H+\epsilon A))^{-1} (H+\epsilon A)^T R^{-1}$

I have seen a kind of contradiction in a paper and I decided to rewrite the equations... Could you please help me to be sure about what I am doing... Let us define $H^\dagger \triangleq ...
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3answers
352 views

Complex Analysis Taylor Series

So the problem states: "Say f(z) := log z is the principal branch of the logarithm (the primitive of 1/z on the region C(-infinity,0]). Show that the Taylor series of f(z) about $z_0 = -1 + i$ takes ...
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83 views

First order multivariate approximation

To demonstrate that $\nabla\!_{\hat{\boldsymbol u}}\,f(\boldsymbol{x}) \equiv \left \langle \hat{\boldsymbol u}, \nabla f(\boldsymbol{x}) \right \rangle$ I plug a first order expansion of ...
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169 views

How to compute the Lagrange remainder of a Taylor expansion

In a Taylor expansion with Lagrange remainder, how can I compute the remainder $R_n(x)$? How to find the $(n+1)$th derivative? Please explain it with elementary functions like $\ln(x),\sin x$ and ...
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1answer
57 views

Talor series of fraction addition of common function

What is the typical trick for finding the taylor series of a common function that is in the denominator when adding a constant. eg: $$f(x)=\frac{1}{e^x-c}$$ I know you can write ...
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521 views

The Idea Behind Taylor Series

I understand that they are viewed as approximations, but was that Taylor's original hope? Assuming that a function can be written as a power series seems to me to be a wild assumption, without some ...
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126 views

About Hessian of distance function

I'm studying the comparison Hessian theorem and I not understand the following: Let $(M, \langle\ ,\ \rangle)$ be a complete Riemannian manifold. Given $o\in M$, define $r=dist(o, \cdot)$. Then, for ...
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66 views

Taylor expansion proof

It's pretty clear to me that in this expansion: $$p(x) = f(0)+f'(0)x+f''(0)\frac{x^2}{2!}+f'''(0)\frac{x^3}{3!}+\cdots$$ When I assume $x=0$, $p(0)$ is gonna be equals to $f(0)$ and all its ...
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What is the Taylor expansion of $W( \vec{n}+ \varepsilon \vec{\eta}, \nabla \vec{n} + \varepsilon \nabla \vec{\eta})$

What would the Taylor expansion of $$W( \vec{n}+ \varepsilon \vec{\eta}, \nabla \vec{n} + \varepsilon \nabla \vec{\eta})$$ be about $\vec{n}$, to order $\varepsilon$? Where $$\nabla ...
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Taylor series Question

So I have a test next week and I saw this question with no answer and I would like to some help. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ infinitely differentiable and let $\sum _{n=0}^{\infty} ...
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1answer
109 views

Use Taylor sequence write approximate value

Use Taylor sequence write approximate value: $$ \sqrt{9.5} $$ Estimate the error approximations three components. Which function should I expand?
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2answers
86 views

Prove that $\sin(x^2) = \mathcal{o}(x)$

I tried to do this like that: $$ \sin(x^2) = \mathcal{o}(x) \iff \lim_{x \to 0} \frac{\sin(x^2)}{x} = 0$$ we could get $\sin(x^2)$ from Taylor series. For $x_0 = 0$, $T_n = 0$ for every $n$. So from ...
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4answers
214 views

Trigonometric Coincidence

I Know that using Taylor Series, the formula of $\sin x$ is $$x-x^3/3!+x^5/5!-x^7/7!\cdots,$$ and the unit of $x$ is radian (where $\pi/2$ is right angle). However, the ratio of the circumference ...
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2answers
55 views

Function and Maclaurin series

Function $f(x)=\frac{x^2+3\cdot\ e^x}{e^{2x}}$ need to be developed in Maclaurin series. I can't find any rule to sum all fractions I've got...so any suggestion that helps? Thanks
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171 views

How does a taylor series of a binomial function equals a trigonometric function? [closed]

Any proof or derivation for the sinx and cosx function would be help. Image taken from http://en.wikipedia.org/wiki/Taylor_series
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2answers
90 views

How can we determine the number of terms which we have to take in a series to get a particular accurate?

As I remember , two days ago , there was a question ( here ) asks for calculating this limit $\displaystyle \lim \limits_{x\rightarrow \infty } \frac{x^3}{e^x}$ and the question was answered . of ...
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1answer
433 views

Why is$ (1+\frac{1}{n})^n=e$ when n goes to infinity? [duplicate]

Why is $\lim\limits_{n\to\infty}(1+\frac1n)^n=e$? I think it involves $\sum\limits_{n=0}^\infty\frac1{k!}=e$ but not sure how to get from one to the other.
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2answers
92 views

Taylor expansion

Is there an easier way to do a Taylor expansion of $e^{u^2+u}$ than do derivatives or substitute and then use Newton's binomial? For example, expanding until the $4$th term: $$e^{u^2+u}=1+u^2+u+ ...
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1answer
166 views

Thinking through a Taylor error bound for arcsine

In lecture, we went through solving a Taylor error bound for arcsine. I followed most of it except for where it talks about the odds divided by the evens divided by $2n+1$ gaining in accuracy by a ...
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92 views

Calculating $\arctan(3)$ using Taylor series

I'm trying to get a Taylor series equivalent for $\arctan(3)$, but the standard definition for $\arctan(x)$ is restricted to $|x| \le 1$. How can I get a Taylor series for this expression?
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derative of Taylor expansion

I'm reading this part of article about key points localization in image processing, and there is something I don't quite understand, mathematically it's this, $$D(w) = D + {\frac{\partial D}{\partial ...
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1answer
338 views

Numerical analysis Taylor's method question: Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$.

Let $f(x)=\tan^{-1}(x)$ Let $P_n(x)$ be the $n$th Taylor polynomial for $f(x)$ about $x_0=0$ Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$. Is ...
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65 views

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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1answer
90 views

Taylor series and its relation to sine

I recently read, one of the most inspiring pieces of literature I've seen, Lockhart's Lament. And now I find myself constantly doing math for fun in my head with imaginary perfect shapes. One such ...
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2answers
147 views

Taylor expansions of $\text{atan}(\tan(x))$ and $\text{asin}(\sin(x))$

Do they actually exist? At least in a form that doesn't degenerate into a mantissa function or into repeated ranges of f(x)=x.
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Taylor Polynomial in Multivariable Case

I am doing Problem 9.30 in Rudin's Principles of Mathematical Analysis and have done part (a) and (b) but got stuck on part(c). In part (b) I have finished the proof that: $$f(\mathbf a+\mathbf ...
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1answer
210 views

How to determine the series for $ f(x) = \sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+x}}}} $ around $0$?

In trying to answer a recent MSE-question I came on the partial problem to determine the power series for the function $$ f(x) = \sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+x}}}} $$ I was not successful ...
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Are Taylor series and power series the same “thing”?

I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general ...
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Taylor expansion with random variables $\frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}$

In Einsteins theory of relativity the kinetic energy of an object is given by the following formula $$E_k = \frac{mc^2}{\sqrt{1-\frac{v}{c}^2}} - mc^2$$ where m is mass of the object at rest v iss ...
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1answer
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trying to understand the incremental form of the Taylor series expansion

original form of Taylor series $f(x)= f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2+ \cdots\\\\$ By making a substitution, we can find something that resembles the incremental form of the ...
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191 views

Taylor Expansion of the 1/2th Derivative

In trying to solve the problem $\sqrt D f(x)=g(x)$ I tried to expand the derivative as a Taylor series, and have encountered a lot of problems. Is there some reason that this shouldn't be possible? ...
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3answers
668 views

Taylor and Maclaurin Series for $f(x)=e^x$

I just came from a final exam where in one question I was asked to derive the Taylor Series for $f(x)=e^{2x}$ centered at $x=1$. I came up with the following: ...
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1answer
75 views

Taylor Expansion with Integral Remainder Question

I have the following question at hand and I have to admit that I am not used to integral remainder form of taylor approximation. I am still trying to work around, so a couple of hints would be useful ...
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1answer
208 views

Is exponential function analytic over all complex numbers

In my textbook, I find a text where it says $e^z$ is analytic everywhere (in complex plane). Is it true? If so, what is the proof? I approached using maclaurin series, which gives $e^z= ...
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1answer
104 views

Series Expansion of an Exponential with a Trig Function in the Exponent

Can anyone get a general expression for $$e^{a\cos x}$$ in terms of an infinite sum? I'm having trouble with a general form in terms of $n$ for the coefficients... Alex
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Question involving approximation, taylor series and proving

Question: Consider the approximation $$\ln(2)\approx 2\left ( \frac{1}{3}+\frac{1}{3\times 3^{3}}+\frac{1}{5\times 3^{5}} \right )$$ Prove that the error in this approximation is less than ...
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Why is the Taylor series $\ln(1-2x) = \sum\limits_{k=1}^\infty (-1)^{k+1}\frac{(-2x)^k}{k} $ incorrect?

We know that: $$\ln(1+x) = \sum\limits_{k=1}^\infty (-1)^{k+1}\frac{x^k}{k} $$ Can we replace $x$ by $-2x$ and get: $$\ln(1-2x) = \sum\limits_{k=1}^\infty (-1)^{k+1}\frac{(-2x)^k}{k} $$ this?
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Find Coefficient in expansion

What will be the coefficient of $x^8$ in the expansion of $x^2\cos x^2$ around $x=0$? I know that it can be done using Leibniz formula for higher order diferentiation of product of two ...
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$\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}x^k =$

Prove that for $|x|<1$, $\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}(-x^2)^k = ...
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Asymptotic expansion of $\ln\left(\frac{x+a}{x-a}\right)$ in form of $\sum\limits_{n=0}^\infty a_n \left(\frac{1}{x}\right)^n$?

How can I find an expansion for $f(x)=\ln\left(\dfrac{x+a}{x-a}\right)$ in terms of powers of $x$, in the form of: $$f(x)=\sum_{n=0}^\infty a_n \left(\frac{1}{x}\right)^n$$ When I try a Taylor ...
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1answer
173 views

Raise a power series to a fractional exponent?

In showing that $\log^\alpha{(1+x)}$ is $O((x)^\alpha)$ at $1$, for $\alpha>0$, one can note that $$\left ( \frac{\log{(1+x)}}{x} \right )^\alpha \overset{x\to 0}{\longrightarrow} \left ( 1\right ...
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Error in approximation of nonzero root to $x^2=\sin x$ using Taylor's cubic polynomial.

I have successfully obtained the root's approximation $r=\sqrt{15}-3$ as I'm supposed to as following:$$\begin{align} \displaystyle f(x)=\sin x &= x - \frac{x^3}{6} + E\\ x^2 &\approx x - ...
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1answer
57 views

Applying the Taylor series

For the initial value problem $\dot{y} = f(y), y(t_0) = y_0$, where $f(y)$ is smooth, we look at the discrete evolution $\Psi^t := y_1 = y_0 + h f((1-\Theta)y_0 + \Theta y_1)$, where $\Theta \in ...
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$\sin(x)$ infinite product [duplicate]

In the equation: $$\sin(x) = x\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)$$ I know that if I do $$0 = x\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)$$ All the roots are really ...
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1answer
76 views

What's Taylor expansion of: $f(x)=\frac 1x\ln{(1+2x^2)}$?

What's Taylor development on the next function: $f(x)=\frac 1x\ln{(1+2x^2)}$? Actually this one is the first question I've seen with $ln$, My instincts tell me to try and do derivative in order to ...
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1answer
97 views

Taylor series of a power series.

Consider a power series $f(x)$ around a point $c \neq 0$. Then is it equal to its Taylor series around $0$? The reason I am wondering about this is because if it is true even for some special cases, ...
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1answer
146 views

Uniqueness of approximations like the Taylor polynomial

Given a function $f: \mathbb {R} ^n \to \mathbb {R} $, I am curious about the uniqueness of a $k$th-order approximation at $c \in \mathbb {R}^n $, i.e. a function $\phi(x)$ such that $$ \frac {f(c ...
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1answer
578 views

how do you determine what the coefficients are on a taylor series expansion if the derivative is too hard to compute?

In a past lecture we talked about how you need to expand The Taylor series of a composed function based on what its input is. For, example: $e^u$ where ${\color{red} u} = \cos x=1 - \frac{1}{2!}x^2 ...