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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Bounding the error term in Taylor's theorem for multivariate function

I have a multivariate function f. I am expanding it according to the Taylor's expansion. I am evaluating it at two points x and y near a fixed point q. I need to evaluate the value of the difference ...
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Approximate $\ln2$ using the Maclaurin series expansion of $\ln\cos x$

Until now I have that $\ln\cos x\approx-\dfrac{x^2}{2}-\dfrac{x^4}{12}$. Since $\cos^{-1}2$ does not exist I do not know what value of $x$ to take. I suppose I need to play around with the original ...
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Are there any entire functions that have a finite number of non-zero terms in their Taylor expansions?

Besides polynomials, of course. I am pretty sure the answer will be "no", but can we actually prove this? Thanks,
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1answer
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How do I derive the Maclaurin series for $\tanh(x)$?

I've thought of doing it by writing $\tanh(x)$ as $(1-e^{-2x})/(1+e^{-2x})$ and then using the Maclaurin series for $e^{x}$ or just as $\sinh(x)/\cosh(x)$ and using the Maclaurin series for $\sinh(x)$ ...
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Help with a Taylor expansion

I want to find the Taylor polynomial $T_{n,f,0} (x)$ of the function: $f(x)=\displaystyle\int_{0}^{x} e^{-t^2}dt$ It cannot be done using the classic calculations, but I may use the theorem of ...
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On a Lagrange remainder application.

Suppose I have $f(x)$ where $f$ is three times continuously differentiable. Then from the Lagrange form of the remainder $ R_k(x) = \frac{f^{(k+1)}(c)}{(k+1)!} (x-x_0)^{k+1} $ I have $$f(x) - ...
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Why doesn't the Taylor series always converge?

The Taylor expansion itself can be derived from mean value theorems which themselves are valid over the entire domain of the function. Then why doesn't the Taylor series converge over the entire ...
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Maclaurin Series of $\ln(2-e^{-x}) $

I tried to solve this by using the series for $e^{-x}$ and $\ln(1+u)$ $$e^{-x}=1-x+\frac{x^2}{2}-\frac{x^3}{6}+...\\ ...
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2answers
44 views

Quick Method to Calculate the Maclaurin Series of $\frac{1}{\sqrt{\cos{x}}} $

I am supposed to calculate the maclaurin series for $\frac{1}{\sqrt{\cos{x}}} $ but I can't seem to figure out an efficient way to go about doing this.
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Maclaurin Series of $\frac{2x}{e^{2x}-1}$ [duplicate]

Calculate the Maclaurin series of $$\frac{2x}{e^{2x}-1} $$ I've tried to calculate it but the series $\frac{1}{e^{2x}-1}$ divides by 0 when x is equal to 0
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2answers
35 views

Which is greater ? Sum of odd power terms or even power terms in the exponential Taylor series?

I came across this question, in a book. Define $f(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{{(2n+1)}!} $ and $ g(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{{(2n)}!} $, where x is a real number. Then, ...
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Taylor Expansion of tensor moved by a flow.

I am reading Peter Petersen's notes on manifold theory and he introduces Lie Derivatives in the following way. "Let $X$ be a vector field and $F^t$ the corresponding locally defined flow on a smooth ...
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1answer
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Taylor expansion at two different points.

My question is stated in the image. I want to prove that every coefficient is same for all terms. But it is not easy. Can you give me some direct proof?
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1answer
63 views

Composition of Pseudodifferential Operators - Remainder term of Asymptotic Expansion

first off this is my first time posting here so I am only learning how to format questions. Please bear with me. I am trying to prove the asymptotic expansion for the symbol of the composition of ...
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Taylor expansion need help understanding.

I am at the moment reading a paper (SURF) and trying to understand what is happening here and how the things works as it does.... a non maximum supression is performed on the scale space ...
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3answers
86 views

The maclaurin series $ f(x) =\frac {x^3} {2+ x^2}$

I know we have exams today and I am doing practise since our lecture; said we need to review our Maclaurin series and I found this question and I wanted to know how one would approach it. Find the ...
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Taylor series of a twice-differentiable scalar function

I've come across this passage somewhere on wikipedia: If $f(t,x)$ is a twice-differentiable scalar function, its expansion in a Taylor series is $$df = \dfrac{\partial f}{\partial t}dt + ...
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2answers
66 views

Taylor series of $[\log(1-z)]^2 $

I'm having some trouble proving that the Taylor series about the origin of the function $[Log(1-z)]^2$ to be $$\sum_{n=1}^\infty \frac{2H_n}{n+1}z^{n+1}$$ where $$H_n = \sum_{j=1}^n \frac{1}{j}$$ So ...
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Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
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The Taylor series of the squared logarithm [duplicate]

Prove that the Taylor series about the origin of the function $[\log(1-z)]^2$ is given by $$\sum_{n=1}^{\infty} \frac{2H_{n}}{n+1} z^{n+1}$$ where $$H_{n} = \sum_{j=1}^{n}\frac{1}{j}$$ is ...
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What is the Taylor series of $e^x$ centred at $3$?

$$ \sum_{k=0}^n \frac{e^3}{n!}(x-3)^n $$ This is my answer - is it correct?
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1answer
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Taylor series at $a = 0$

If we are given a power series with $a = 0$ and it converges to $f$ in some interval around $a$, then the power series is the taylor series to $f$. But what is the taylor polynomial of $f$ to some ...
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Proof of a step of a lemma on the asymptotics of maximum likelihood where a Taylor expansion is used. (crosspost from crossvalidated).

I have asked this question on crossvalidated here and I am still unsure on the answer. I attempt a cross-post (most of the times this proves very useful). I copy the question below: I am trying to ...
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Approximation formula for third derivative, is my approach right?

Derive by using Taylor approximation up to 4th degree (in $h$) of $f$ in $x_0 \pm h$, $x_0\pm 2h$ at $x_0$, an formula for approximation of $f'''(x_0)$ with an error term of order $h^2$. Could ...
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1answer
123 views

The power series $\sum_{n\geq 1} \frac{x^{n}}{n(2n-1)}$ with $2$nd Taylor polynomial and Taylor series. [Solved]

I have been a fool not noticing it earlier. Instead of deleting this thread I have chosen to put the short solutions of this problem. This thread is closed. Consider the series $$\sum_{n\geq 1} ...
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Taylor series for a multivariable function

We know the following: $a \approx 1 + x\sqrt{dt}$ $V_1 = V(aS, t + dt ) $ The textbook claims you can (using Taylor's Theorem), expand the bottom-most equation like this: $V_1 \approx V + ...
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3answers
243 views

Rewrite trigonometric expression to be be numerically “stable”

Is it possible to write the following function: $$ f(x) = \begin{cases} \frac{x-\sin x}{1- \cos x}& x\neq 0\\ 0 & x=0 \end{cases} $$ as a composition of elementary functions (including ...
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2answers
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Find a formula for the binomial coefficients of the Macluarin series for $\frac{1}{(1+x)^{1/2}}$

The Maclaurin series for $\frac{1}{(1+x)^{1/2}}$ is \begin{equation*} 1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\frac{35x^4}{128}...~. \end{equation*} I can't figure much out other than it ...
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How to find the largest positive real value r for Maclauren polynomials where error is set

I need some tips on answering this question: The function is $\sin^2 (r)$ Using Taylor's Theorem, determine the largest positive real value r for which we can guarantee that the Maclauren ...
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2answers
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Taylor expansion of a complex function

Trying to find Taylor series of $$\frac{z^2}{(1+z)^2}$$ I write it in the form $1- \frac{2}{1+z} + \frac{1}{(1+z)^2}$ and I can find Taylor expansion for each factor, is there another method without ...
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Taylor series of $1\over z^2$

How to find the Taylor series of $1\over z^2$ near $2$ ( in the power of $z-2$) I have tried to write it in the form: $1\over ((z-2)^2+4z-4)$ But I reached nothing, any help please
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Taylor series in complex analysis [closed]

I am working on finding the Taylor series of $$\frac1{az+b}$$ in powers of $z.$ How to start with it Any help in details...
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Theoretical Question regarding Taylor Expansion

I received the following question during a Calculus $2.0$ course in my university. I am not a native speaker, so please excuse my English. The question is as follows: Let $f$ be a function with ...
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Laurent series for $\frac{2}{(z)(z-1)(z-2)}$

! So I think I am getting the hang of Laurent Series, but having a bit of trouble with one of the fractions for part a). So I split this up in to partial fractions: $\frac{1}{z} - ...
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Explanation of taylor series

I understand that for a Taylor series of a function $f(x)$, centered around the point a, the general expression can be written as: $$ \begin{align} &f(x) \\ &= f(a) + f'(a) (x-a) + ...
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Find a power series that will converge to F(x)

The question is: Find a power series that will converge to $F(x) = \int_0^x\sin(t^2)\;dt$. I don't really have any idea how to solve this, but I know that I need to create the Maclaurin expansion so ...
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Is there an alternative to Taylor expansion of functions with more control over the error distribution?

As a physics student I see the Taylor series being (ab)used very often, mostly for the expansion of $\exp(x)$. The usual line of though is that the error (remainder) term of a high order is likely ...
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Find Taylor-Maclaurine expansion of function

Find Taylor-Maclaurine expansion of function: $f(x)=\sin(x)\cdot \cos(x) \cdot \arctan x^2$ my try: $f(x)=\frac{1}{2}\sin{2x} \cdot \arctan x^2$ and we have $\displaystyle \sin{2x} = \sum _{n=0} ...
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Solve $x^2=\cos x$ using Taylor series for cosx

I have the following equation:$x^2=\cos x$ and calculating the Taylor series of $3rd$ degree around $0$ I've got: $x\approx \pm\sqrt{\frac{2}{3}}$ However, now I need to prove that if x is a ...
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3answers
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Taylor series $\ln(1+e^x)$ about $x=0$

What's the best way to determine this up $x^3$ terms? I thought it would be to take the series for $\ln(1+x)$ and the series for $e^x$ up to $x^3$ and sub the second series into the first. ...
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1answer
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Upper bound for truncated taylor series

A paper claimed the following but I can't figure out why it's true: For all $1/2> \delta > 0$, $k\le n^{1/2-\delta}$, and $j\le k-1$ where $n$, $j$, and $k$ are positive integers, the following ...
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Taylor series in two variables?

how can I calculate the taylor series for a two-dimensional function? Example: \begin{equation*} f(x,y) = Log(1+x+y). \end{equation*} I have $f_x = (1+x+y)^{-1} = f_y$. $f_{xx} = -(1+x+y)^{-2} = ...
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1answer
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Taylor Expansion Derivation

I have a (perhaps silly) question, but I am reviewing Taylor series approximation and looking at this slide. In the third panel, $f(a+h)$ is isolated but I am confused why $O(h^{2})$ does not have a ...
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1answer
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Help Regarding The Taylor Series Remainder Proof Understanding

I'm reading Mathematics: It's Content, Methods, and Meanings and I am in a chapter about Taylor Series. It made sense until I came across the remainder part of the theorem. In order to prove the ...
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Need help with taylor series.

Evaluate the limit $$\lim\limits_{x \to 1} \frac{1-x + \ln x}{1+ \cos πx}$$ The limit im trying to get is $-\frac{1}{π^2}$ as I've solved from l'Hopitals rule. Now I need to solve the limit by ...
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2answers
43 views

When is Taylor series substitution valid?

Given that the Maclaurin series for $g(x) = \frac{1}{1+x}$ is $1 - x + x^2 - x^3 + x^4 ... $, I'm told that the Maclaurin series for $\frac{1}{1+x^2}$ is $1 - x^2 + x^4 - x^6 ... $, by substituting in ...
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Write down the Maclaurin series for the function.

(d) Another function is defined on the interval of convergence of the power series in (b) by the formula $\displaystyle{g(x)=\int_0^x \ln(2t+1) dt}$. Write down the Maclaurin series for the ...
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1answer
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cubic approximation with four points(approximating sine function with polynomials)

I was reading the following article https://mixedmath.wordpress.com/2013/11/17/an-intuitive-overview-of-taylor-series/ regarding Taylor Series.When I got to the part 1.3. Cubic approximation I got ...
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44 views

Why the general formula of Taylor series for $ln(x)$ does not work for $n=0$?

I need to find the taylor series for $log(x)$ about $a = 2$, and I have find the following solution, but I don't understand why the general formula does not work for $n = 0$.
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Maclaurin series of $y=\ln (\dfrac{1+e^{-x}}{2})$ with $\dfrac{dy}{dx}=\dfrac{e^{-y}}{2}-1$

Maclaurin series of $y=\ln (\dfrac{1+e^{-x}}{2})$ with $\dfrac{dy}{dx}=\dfrac{e^{-y}}{2}-1$ This question requires you to use the given result of $dy/dx$ I've worked out that ...