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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2
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1answer
35 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 = ...
2
votes
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 ...
0
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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?
0
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1answer
19 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. ...
0
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0answers
17 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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0answers
20 views

Help me understand procedure of integrals. [on hold]

Help me to calculate: $\int \sin (-x^2 )dx$ approximation by Taylor 2. series for $x_0=0$. Thank you
-1
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1answer
27 views

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 ...
0
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1answer
41 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 ...
2
votes
1answer
33 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 ...
1
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1answer
43 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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0answers
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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 ...
2
votes
1answer
34 views

To find value of n using taylor series expansion [on hold]

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 ...
0
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0answers
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.
1
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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) → ...
6
votes
1answer
127 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 ...
1
vote
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 + ...
3
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2answers
33 views

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} + + ...
0
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0answers
15 views

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 $R(x, y) = f(x, y) - P(x, y)$ ...
2
votes
1answer
20 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 ...
1
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1answer
33 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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0answers
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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vote
2answers
27 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
19 views

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. ...
1
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1answer
46 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?
2
votes
3answers
36 views

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 ...
3
votes
1answer
15 views

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 ...
0
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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 ...
0
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0answers
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 ...
3
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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 ...
1
vote
2answers
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 ...
1
vote
1answer
33 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
37 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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0answers
29 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
62 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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votes
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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vote
1answer
25 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 ...
1
vote
1answer
40 views

Taylor series expansion of $e^{x+y}$ about the point $(0,1)$

My question is: what is the Taylor series expansion of $e^{x+y}$ about the point $(0,1)$? I think the standard $e^{x+y} = 1 + x+y + 1/2(x+y)^2$ ... doesn't apply here. Thanks in advance
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votes
2answers
56 views

Upper Bound for $|f^{n}(0)|$ given that $f$ is Analytic

Let $f(x)$ be an analytic function in some neighborhood of $x=0$. $f$ being analytic implies that its has a convergent Taylor series expansion about $x=0$. That is, there exists $R>0$ (radius of ...
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votes
7answers
207 views

Taylor series for $\sqrt{x}$?

I'm trying to figure Taylor series for $\sqrt{x}$. Unfortunately all web pages and books show examples for $\sqrt{x+1}$. Is there any particual reason no one shows Taylor series for exactly ...
1
vote
2answers
33 views

Taylor Series approximation

Let $f(x) = (1-x)^{-1}$ and $x_0=0$. Find the $n$-th Taylor polynomial $P_n(x)$ for $f(x)$ about $x_0$. Find a value of $n$ necessary to approximate $f(x)$ within $10^{-6}$ on $[0,0.5]$. I am ...
0
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1answer
58 views

Taylor series of a taylor series: why can we do this?

Suppose we have two functions with the following Taylor series: $$sin(x) = x-x^3/3!+x^5/5!+O(x^6)$$ $$e^x = 1+x+x^2/2+x^3/3!+x^4/4!+x^5/5!+O(x^6)$$ I know, by intuition and because that's what we got ...
1
vote
1answer
47 views

Find the residue at $z=-2$ for $g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$

Find the residue at $z=-2$ for $$g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ I know that: $$\psi(z+1) = -\gamma - \sum_{k=1}^{\infty} (-1)^k\zeta(k+1)z^k$$ Let $z \to -1 - z$ to get: $$\psi(-z) = ...
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votes
2answers
46 views

Find the McLaurin series for $f(x)=x/(x+1)$

Can you help me find the McLaurin of $f(x)=x/(x+1)$ ? I am new to this mathematical chapter and already tried but I do not think my result is correct.
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vote
1answer
42 views

Taylor expansion about a point

I need help with the following calculus problem: Use completing the square and the geometric series to get the Taylor expansion about ${x=2}$ of ${\frac{1}{x^{2}+4x+3}}$ So far I have the following: ...
2
votes
2answers
37 views

Reverse engineering a Taylor expansion

We have the sum: $$S(x) = \frac{x^4}{3(0!)} + \frac{x^5}{4(1!)} + \text{ }...$$ And we are told to sum the series to obtain a finite expression. My guess was to reverse engineer the expression in ...
6
votes
1answer
32 views

the Zassenhaus /Baker–Campbell–Hausdorff formula for cosine.

This question concerns the expansion of non-commutative algebra $[X,Y] \neq 0$ for two operators $X,Y$. One can think of $X$ and $Y$ as some matrices. If $[X,Y] = 0$, we have $$e^{t(X+Y)}= e^{tX}~ ...
0
votes
1answer
39 views

Taylor series expansion in calculus of variations

I am reading a book on calculus of variations, so I stumbled upon this integral, which the author expands by taylor series expansion, where $y$ and $y'$ are functions of $x$ and $\tilde{y}(x) = y(x) ...
0
votes
0answers
19 views

Discretisation of a product of two functions

Suppose I have two functions, $f(x,t)$ and $g(x,t)$, and for an upwind scheme I want to use the quantity $\partial_x (fg)$ to solve the advection equation $$ \frac{\partial f}{\partial t} + ...
1
vote
0answers
21 views

Prove that $f$ has a global maximum given a product of Taylor polinomials

Let $f:\mathbb{R}\to\mathbb{R}:f\in C^{\infty},f\left(1\right)=0$ and the product of its Taylor polinomials of order 2 in $x_{0}=0$ and $x_{0}=1$ be $P(x)=x^{4}-2x^{3}+2x^{2}-x$. Prove that $f$ has a ...
0
votes
1answer
26 views

Taylor expansion of $f(z)=\frac{z-1}{z^2-3z+3}$

We are given the function $f: \mathbb C \to \mathbb C$ defined by $f(z)=\frac{z-1}{z^2-3z+3}$ Is it possible to define $f$ as its taylor expansion near the point $z=i\sqrt 3$? If so, what is the ...