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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MacLaurin Series with Variable in Denominator

A friend of mine was talking about how finding MacLaurin series for functions with variables in the denominator might prove difficult without tables. We started making lots of crazy problems, but one ...
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2answers
16 views

Poisson complete statistic

I have the same question as this thread, but I cannot understand the proof. The problem is, given $f(\lambda)=\sum_{k=0}^\infty g(k)\frac{(n\lambda)^k}{k!}=0,\forall\lambda>0$. How to show ...
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4answers
67 views

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$.

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. I can't even start because I don't really know what $x\to 1^-$ means. If you know what it means it would really help me. I would as well if you ...
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2answers
83 views

How to calculate $\exp(-x)$ using Taylor series

We know that the Taylor series expansion of $e^x$ is \begin{equation} e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}. \end{equation} If I have to use this formula to evaluate $e^{-20}$, how ...
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0answers
17 views

Coefficient in Taylor Series expansion [duplicate]

Find the coefficient of $(z-\pi)^2$ in the Taylor series expansion around $\pi$ if $$f(z) = \begin{cases} \frac{\sin z}{z-\pi} & \quad, z \neq \pi \\ -1 & \quad, z=\pi \end{cases}$$
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1answer
19 views

Number of derivatives in a taylor series expansion

I would like to confirm if the number of derivatives we need to calculate in a specific order of a taylor series expansion is the sum of the multinomial coefficient of that order: $$ f:\mathbb{R}^k ...
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1answer
58 views

Evaluating $\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k} $

Question: How to compute $$ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k}? $$ Here is what I have tried so far: Define $s_n=\sum\limits_{k=1}^n \frac{1}{k 2^k}$ for every index $n$, ...
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1answer
17 views

Taylor theorem for f(x+h)

I am following a proof that applies Taylor's theorem on this document (http://www.gautampendse.com/software/lasso/webpage/pendseLassoShooting.pdf) I am not understanding one of the terms explained on ...
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14 views

A simple question about Delta Method's demonstration.

Suppose that $\sqrt{n}(X_n-\mu)\stackrel{D}{\longrightarrow}X$ and consider $g:\mathbb{R}\rightarrow\mathbb{R}$ a function such that first derivative $g'$ is continuous in a neighbourhood of $\mu$, ...
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3answers
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taylor of $\frac{1}{z}$ at $a=-2$

I want to find the taylor series representation of $f(z)=\frac{1}{z}$ at $a=-2$. The point of this exercise is not to find some pattern in the derivatives, infact we are not meant to find any ...
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4answers
94 views

Find $\lim_{n\to \infty}(\cos{x\over n})^{n^2}$

Find $$\lim_{n\to \infty}\left(\cos{x\over n}\right)^{n^2}$$ where $x\in \Bbb{R}.$ I tried using taylor series. A complete mess, and an area I am not very good at. I tried using $e$ which also gave me ...
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1answer
30 views

Expansion $ f (x) = \ln^ 2 x$. Taylor. [on hold]

Expand the function $ f (x) = \ln^ 2 x $ in a Taylor series at the point $ x_0 = 1 $ Please help me do it.
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1answer
17 views

nth derivative of a troublesome function

I don't know where to start on this problem. I'm trying to get the 2015th derivative(at x = 0) of f(x) = x^2 * arctan(x). Doing the derivatives one by one seems a little troublesome... What do you ...
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2answers
42 views

What is the coefficient of $(z-\pi)^2$ in Taylor series expansion of $\sin (z)/ (z-\pi)$

I want to determine the coefficient of $(z- \pi)^2$ in Taylor series expansion of $f(z)=\sin (z)/ (z-\pi)$ if $z \neq \pi $, $-1$ if $z=\pi$ around $\pi$. How can this be done? I don't know how to do ...
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1answer
38 views

Sum of exponential square series

I have a infinite sum which I wonder if it will converge to a simpler function $f(r) = \Sigma_n r^{n^2} , r<1$, I also interested in case $r$ is a complex number on unity circle $r = ...
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1answer
35 views

Higher Order Terms in Stirling's Approximation

Some websites and books give stirling approximation as $$n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \left( 1 + O \left(\frac{1}{n} \right)\right)$$ However when I check their derivations most ...
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1answer
38 views

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$ Should I use the formula $C(n,k) = n!/[k!(n-k)!]$? And what is the solution of this problem?
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1answer
21 views

taylor series approximation of e function

in the equation $$e^{y(x)}=1+2x-\frac{y(x)}{1-x}$$ $y(0)=0$ because using the taylor series and by comparing the coefficients we obtain $$1+y(0)=1-y(0)$$But why is using the taylor series allowed. ...
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1answer
26 views

Functionals' Taylors Theorem

Consider functional $F:B\to \mathbb{R}$, where B is a Banach space eg. $B=H^{1}(\mathbb{R}^{d},\mathbb{C})$. Then Taylor's theorem for functionals is: Suppose that the line segment between u ∈ ...
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Help me understand procedure of integrals. [closed]

Help me to calculate: $\int \sin (-x^2 )dx$ approximation by Taylor 2. series for $x_0=0$. Thank you
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1answer
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Maclaurin Expansion of $\ln(1+4x^2+4x)$ in terms of $\sum a_k x^k$

Maclaurin Expansion of $\ln(1+4x^2+4x)$ in terms of $\sum a_k x^k$ The question has written to $x^2$ term before the $x$ - does that have anything to do with how to solve the problem? Am I meant to ...
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1answer
44 views

Taylor expansion in $p$-adic integers

Let $f \in Z_p[X]$, then for $ x, y \in Z_p$, $\exists a \in Z_p$ s.t. $f(y)=f(x)+(y-x)f'(x)+(y-x)^2a$. Why is Taylor formula applicable to polynomial in $p$-adic integers $Z_p$? What condition ...
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1answer
34 views

Extending Taylor's theorem from one to several variables

In my calculus class we are dealing with Taylor´s theorem in several variables. When we were looking at the function $f(x,y)=\sin(xy)$ my teacher said that instead of applying the theorem in several ...
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1answer
47 views

Using $\ln (\cos x)=\frac{-x^2}{2}-\frac{x^4}{12}+…$, approximate $\ln 2$ in terms of $\pi$

Using $f(x)=\ln (\cos x)=\dfrac{-x^2}{2}-\dfrac{x^4}{12}+\dots $, approximate $\ln 2$ in terms of $\pi$. I know $\cos(x)$ will never be two - so what can I actually substitute in to get something ...
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second derivative fail, classify the nature of the critical point

f(x,y,z)= $\frac{(x+y)^2}{2}+z^3$ the critical point I calculated is span{(1,-1,0)} the eigenvalue of the Hessian of point (1,-1,0) are 0,0,2, which means that this point is degenerate and the ...
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1answer
39 views

To find value of n using taylor series expansion [closed]

Let $$ f(x)=\begin{cases} 0& -1\le x\le0\\ x^4& 0\lt x\le 1\end{cases} $$ IF$$ f(x)=\sum_0^n\frac{f^{(n)}(0)}{n!}(x)^n + \frac{f^{(n+1)}(c)}{n+1!}(x)^{(n+1)}$$ is the Taylor's formula for ...
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15 views

Taylor polynomial to find an approximation

Use the Taylor polynomial of degree 5 to give an approximation for ln(2) This may seem really simple but I have no idea how to do it, please help.
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1answer
27 views

Is the taylor polynomial of degree $2$ near $(0,0)$ of $𝑓(𝑥, 𝑦) = \frac{1}{ 2 - (𝑥 + 𝑦^2)}$ the following:

$ P(𝑥, 𝑦) = \frac{1}{2} + \frac{𝑥}{4} + \frac{𝑥^2}{4} + \frac{𝑦^2}{2}$ Is this right? I can't tell, as I can't seem to see the remainder going to $0$ when divided by $x^2 + y^2$ as $(x, y) → ...
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1answer
129 views

Can $f(g(x))$ be a polynomial?

Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions. Is it possible that $f(g(x))$ is equal to a polynomial ? edit Some comments : I was thinking about iterations. So for instance ...
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1answer
25 views

Functional expansion

I am confused by this expansion in Landau and Lifshitz: First, they define $\textbf{v}' = \textbf{v} + \textbf{$\epsilon$}$. So for a function $L$, $$L(v'^2) = L(v^2 + ...
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2answers
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Another question about $x_0$ in the Taylor series

When we talk about Taylor series, we say it's around point $x_0$. It's in the Taylor series formula: $$f(x) = f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2f''(x_0)}{2} + \frac{(x-x_0)^3f'''(x_0)}{6} + + ...
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0answers
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How can I show the remainder of this Taylor polynomial $R(h)/h^2$ goes to $0$ as $h$ goes to $0$?

Given the function $f(x, y) = 1/(2 - x - y^2)$ I found that the second-degree Taylor polynomial is $P(x, y) = 1/2 + x/4 + x^2/4 + y^2/2$ How can I show the remainder $R(x, y) = f(x, y) - P(x, y)$ ...
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1answer
24 views

Taylor series expansion of the function $f(x)=x \arctan x-0.5 \log(1+x^2)$ about the origin int the region {$|x|\le1$}

Find the Taylor series expansion of the function $\color {green}{f(x)=x \tan^{-1} x-0.5 \log(1+x^2)}$ about the origin int the region {$|x|\le1$} My effort: I know $\displaystyle \log ...
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1answer
34 views

Physically, what meaning have Taylor series which have their lower order terms equal to zero, but their higher order terms non zero?

Usually, when using a Taylor series to describe a function (which may itself be a model of some physical phenomenon), we often throw out the higher order terms, as they are quite small relative to the ...
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34 views

Finding the derivative of the remainder term in a mean value expansion

Consider the following expansion: \begin{equation} f(x)=f(x_0)+f'(x_0)(x-x_0) + \frac{1}{2}f''(\tilde{x})(x-x_0)^2, \end{equation} where $\tilde{x}\in\{\min[x,x_0],\max[x,x_0] \}$. Can we say anything ...
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2answers
31 views

On finding the order of an infinitely small quantity

Given an infinitely small quantity: $$\alpha \left ( x \right )= \tan \left ( x \right )-\sin \left( x \right)$$ as x aproaches $0$, and computing the corresponding asymptotic relationship. What does ...
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0answers
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Name of method which includes Taylor linearization inside fixed point iteration

I read paper about Horn-Schunck multiscale method for computing optical flow Core part of this algorithm is minimizing some functional. One part of functional contains nonlinear term inside L2 norm. ...
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1answer
47 views

Taylor expansion of $1/(1+z)$

How do I obtain the Taylor expansion of $$\frac{1}{1+z}$$ about $a=i$ please? Do I just expand the series using the binomial expansion?
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3answers
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Taylor series of $\ln(1+x)$

So let's say we want to obtain the Taylor series for $\ln(1+x)$. We know that its derivative is $\dfrac{1}{1+x}$, which has the series $\sum_{n=0}^{\infty} (-1)^nx^n$. Can we take the antiderivative ...
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1answer
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using Taylor's Theorem to find region of convergence of series

!(http://imgur.com/0fDL4KZ) I am a third year Electrical engineering student, and I was going through one of the example from my math module lecture notes but couldn't understand the solution printed ...
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1answer
23 views

Find a polynomial $Q$ of degree $k$ and a remainder function $E$ for $f(x)=\frac{1}{1-x}$.

There is a theorem in our textbook saying that rather than calculating all the derivatives needed to compute the taylor polynomial, if one can find, by any means, a polynomial $Q$ of degree $k$ such ...
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25 views

complex taylor series

I have a series $ f(z)= 1 - z + z^2 - z^3 $ and i want to substitute $ z=b + (z-b) $ into the equation, (where $b=1/2+i/2$) and find the first two coefficients. Wont they just be the same as before ...
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0answers
51 views

Prove that the limit is equal to one. [duplicate]

How to prove that $$\lim_{x\rightarrow 0 }\frac{\sin\big(\tan(x)\big)- \tan\big(\sin(x)\big)}{\arcsin\big(\arctan(x)\big)-\arctan\big(\arcsin(x)\big) } = 1$$ First terms in Taylor expansion of the ...
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66 views

How can I prove that $x-{x^2\over2}<\ln(1+x)$

How can I prove that $$\displaystyle x-\frac {x^2} 2 < \ln(1+x)$$ for any $x>0$ I think it's somehow related to Taylor expansion of natural logarithm, when: $$\displaystyle ...
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1answer
35 views

What is the series to converge with $1/x$ from $(1,\infty)$?

I'm trying to find an alternative series of polynomials that can pssibly converge with $\frac{1}{x}$. So far I know that the taylor series for $\frac{1}{x}$ is, as should be known, ...
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0answers
40 views

Partial sum formula for $\sum_{n=0}^{x} {\tan(x)}$ from $(0,\infty)$

I know there is no elementary way of expressing the partial sums of $\tan(x)$. I know; however, I can get an approximation of partial sums using a series, such as the MacLaurin series. If a series can ...
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30 views

Explicit analytic continuation of the zeta function and shift operators

Is there a way to compute the radius of convergence of the expansion of the zeta function, e.g. around $a=2+2i$? We have $\zeta(a)=\sum_{n=1}^\infty n^{-a}\approx 0.867.. - i\,0.275.. $, but I ...
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5answers
63 views

Approximate $\coth(x)$ around $x = 0$

I'm trying to approximate $\coth(x)$ around $x = 0$, up to say, third order in $x$. Now obviously a simple taylor expansion doesn't work, as it diverges around $x = 0$. I'm not quite sure how to ...
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1answer
20 views

Taylor expansion for two-variable function.

Expand the function $ f (x, y) = e ^ {x-2y} $ in a Taylor series at the point $ (- 1,2) $. Please help me with it. I don't know how to do it although I did try to do it.
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1answer
26 views

Finding the limit of a function with sines and cosines by using the taylor expansion

I need to find the residue of a second order pole $z=0$, the residue works out to the following: $$\lim_{z\to 0}\frac{2z\sin{z^2}-2z^3\cos{z^2}}{\text{sin}^2{z^2}}$$ My professor said it's ...