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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Expand a function in Maclaurin's series.

The function is given with: $$\ln(5\cos^{3}(x))$$ Need to be expanded: $$x^{4}$$ I have no idea what to do here and honestly, I don't really understand what a Maclaurin's series is. I know the ...
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2answers
51 views

function approximation using series

I was trying to approximate a function by another one using some kind of a series. Let $$ f(x) = m\cdot \left(1-\sqrt{\frac x2\biggm/\left(1+\frac x2\right)}\right) $$ I'm trying to approximate ...
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1answer
27 views

Taylor Series for $e^x(x^2 -x + 1)$

Find the Taylor Series for $e^x(x^2 -x + 1)$ about $x=0$. More importantly, find the COEFFICIENT (for nonzero terms) of the taylor series. The answer says: $$e^x(x^2 -x + 1) = 1 + ...
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1answer
19 views

For small x, Taylor Series to determine constants n and C in the approximation

cos(x) - e^(-(x^2)/2) I have tried writing the taylor expansion for cosine and e but do not know how to combine them in one. In addition to this, i do not know how to approach the problem whilst ...
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2answers
52 views

Taylor series of $x/(x^2-4x+5)$

I'm supposed to find the Taylor series of this function (I can choose to center it at any A I want): $$f(x)= x/(x^2-4x+5)$$ When I derivate, it only gets more and more confusing. How can I make any ...
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1answer
35 views

Why can you use the Maclaurin Series for certain cases of function not about 0?

Is it possible to use the Maclaurin Series in a problem like this one (AP Calculus BC Question 6 from a few years ago)? Write the first four nonzero terms and the general term of the Taylor ...
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0answers
67 views

Beyond taylor series?

Consider functions $f(z)$ that are analytic for $Re(z) > 0$ and are also analytic for $(Im(z))^2 > 0$. Let $n$ be a nonnegative integer. Now I define some series expansion of "order $n$" , ...
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1answer
50 views

Convenient notation, or something more?

A little while ago I happened across a curious formula that blew my mind (no idea what it's called): $e^{\frac{d}{dx}}f(x)=f(x+1)$ I played around with it a bit and managed to prove it using the ...
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1answer
31 views

Taylor Series to the Power 1/z

I am attempting to find the Taylor Series for $(\frac{\sin{z}}{z})^{\frac{1}{z^2}}$. While I can plug this into Wolfram and use the output, I want to understand how to calculate the Taylor Series ...
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1answer
38 views

Messing with the linear approximation… I don't get Taylor's formula??

Alright, so I was messing around with the linear approximation for a function; I wanted to see if I could get the formula for the 2nd degree approximation from it. I went about it like this: $$ f(x + ...
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2answers
22 views

Taylor Series with differentiator operator

Hi guys can anyone show me how the Taylor series can be converted from: $$f(x+h)= f(x)+hf'(x)+...$$ to: $f(x+h)=e^{hD}f(x)$, where $D$ is the differentiation operator. How does the differentiator ...
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17 views

A formal solution using Taylor series

Consider the following: $S_r(n)= 1^r+2^r+...+(n-1)^r$ where $S_r(n)$ satisfies: $S_r(n+1)-S_r(n+1)=n^r$ Now, also consider the Taylor series $f(x+h)=f(x)+hf'(x)+(h^2/2!)f'(x)+..$ which can be ...
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2answers
19 views

expand function, taylors series, combinatorics, generation functions

I have to expand $f(z)$ into a formal power series $f(z) = \sum\limits_{k=0}^\infty a_kz^k$ (for $z$ close to 0) $f(z)= \frac{z^3}{1-4z+3z^2}$ I know that: $\frac{1}{1-z} = \sum\limits_{k=0}^\infty ...
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1answer
14 views

Equality form of second order Taylor series

I am reading a book on optimization wherein a statement using Taylor's expansion is made without proof. \begin{equation} f(\mathbf{y}) = f(\mathbf{x}) + (\mathbf{y} - \mathbf{x})^T\nabla ...
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1answer
39 views

Need clarification on a Taylor polynomial question

$$f(x) = 5 \ln(x)-x$$ second Taylor polynomial centered around $b=1$ is $-1 + 4(x-1) - (5/2)(x-1)^2$ let $a$ be a real number : $0 < a < 1$ let $J$ be closed interval $[1-a, 1+a]$ find upper ...
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1answer
248 views

Erroneously Finding the Lagrange Error Bound

Consider $f(x) = \sin(5x + \pi/4)$ and let $P(x)$ be the third-degree Taylor polynomial for $f$ about $0$. I am asked to find the Lagrange error bound to show that $|(f(1/10) - P(1/10))| < 1/100$. ...
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1answer
58 views

first order approximation of scalar function of matrix ( Mahalanobis distance)

I have tried to compute the 1st order approximation using Taylor's expansion of the Mahalanobis distance: $f(\mathbf{X})=\mathbf{a^TXa}$, where $\mathbf{a}\in \mathbb{R}^N$. The function maps ...
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1answer
53 views

Complex Taylor Series Circles of Convergence

I am trying to find the Taylor Series and circles of convergence for three different functions. i) $\frac{\sin{z}}{z}$ which I determined the Taylor series to be $\sum_{n=0}^\infty ...
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1answer
32 views

What does the h mean within the Taylor expansion of $y(x_0 +h)$ and $y(x_0 -h)$?

I understand that the Taylor series formula is $$\frac{f^n(a)}{n!}(x-a)^n.$$ I also know that the Taylor series expansion of $$y(x_0 +h)=y(x_0) +hy'(x_0)+\frac{h^2}{2!}y''(x_0)+ ...
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1answer
29 views

Find Taylor Polynomial of degree $2$ about (2,1) where $f(x,y) = x^2y^3$ , $(x,y) \in \mathbb{R}$

Find Taylor Polynomial of degree $2$ about $(2,1)$ where $f(x,y) = x^2y^3$ $(x,y) \in \mathbb{R}$ My thoughts: $D_xf= 2xy^3$ and $D_{xx}f= 2y^3$ $D_yf= x^23y^2$ and $D_{yy}f= x^26y$ $D_{xy} = ...
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1answer
73 views

Gibbs Phenomenon and Fourier Series

a) Show the partial sum $$S = \frac{4}{\pi} \sum_{n=1}^N \frac{\sin((2n-1)t)}{2n-1}$$ which may also be written as $$ \frac{2}{\pi}\int_0^x\frac{\sin(2Nt)}{\sin(t)}dt$$ has extrema at $x= ...
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1answer
57 views

Using Taylor's Theorem and the Constancy Theorem, solve the following proof.

Using Taylor's Theorem and the Constancy Theorem prove that $\sqrt{1+x}=1+\frac{x}{2}+\sum_{n=2}^{\infty}(-1)^{n-1} \frac{1}{2n} \frac{(1- \frac{1}{2})(2- \frac{1}{2}) ... ((n-1)- ...
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1answer
98 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
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0answers
41 views

Substituting for the variable in a standard taylor series

I'm trying to show that 4/(3x+1)=1/(1+(3/4)(x-1)) so i can then find the taylor series for the function f(x)=4/3x+1 and determne an interval of validity I know 4/3x+1 is similar to 1/(1-x) for which ...
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0answers
22 views

Function approximation by various means

I know several ways to approximate a function: Taylor series, Fourier series, or polynomials, like e.g. Legendre polynomials. Is the only difference between those various methods the speed at which ...
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1answer
38 views

Laurent Series - when do singularities on the boundary of an annulus require a Laurent series instead of Taylor?

I need to find the Laurent Expansion of $F(z) = \dfrac{1}{(z-1)^2(z+2)}$ in the regions $A_1 = D(0,1)$ and $A_2 = \{z: 1 < |z| < 2 \}$. After doing partial fractions on $F(z)$, how do I know ...
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4answers
52 views

Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$.

Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$. Taylor's Theorem applies at the point $a=0$ and with $n=4$. Got no idea how to proceed. My lecture notes have one example ...
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2answers
43 views

Trying to determine the Inverse function of $\sinh$ and $\cosh$

I'm trying to find out how to determine the inverse function in order to develop the $$ \sinh(x).$$ I tried to expand to its exponential form $$\sinh(x) = \frac{1}{2} (e^x-e^{-x}) .$$ So I wrote $$ ...
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2answers
152 views

Approximating $\ln(1+\exp(x)+\exp(y))$

Ok, so I know that $\ln(1+e^x)\approx x$ when $x$ is large. But what about $\ln(1+e^x+e^y)$ when both $x$ and $y$ are large? I can figure out cases when $x\gg y$ or $y\gg x$ since that simplifies ...
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1answer
26 views

Is it Possible to Develop an inverse function using the function it self

Is it Possible to Develop (taylor expansion) of an inverse function by knowing the function it self ? If Yes ,Can you illustrate with a simple function I know that we use the identity formula $$ ...
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3answers
900 views

In Taylor series, what's the significance of choosing the point of expansion $x=a$?

So I read about the Taylor series and it said you can choose to expand the series around a given point ($x=a$). Does it matter which point you choose in calculating the value of the series? For ...
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53 views

Estimate ln(3) using Taylor Expansion up to 3rd order

Estimate ln(3) using Taylor Expansion up to 3rd order (without the use of a calculator). $$f(x)=ln(x)$$ $$f'(x)=1/x$$ $$f''(x)=-1/x^2$$ $$f'''(x)=2/x^3$$ ...
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0answers
23 views

Multivariable Taylor expansion

How can I compute $\dfrac{\partial^{4n}}{\partial{x^{2n}}\partial{y^{2n}}}\Bigl(\dfrac{1}{1+x^2+y^2}\Bigr)\Biggl|_{(0,0)}$ ? I guess that Taylor expansion of the function should be used. How?
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30 views

An application of Rouches Theorem

Let $f$ be an entire function on the complex plane, with Taylor's expansion around zero as $f(z) = \sum_{k=0}^{\infty}c_{k}z^{k}$. Let $N(r)$ be the number of zeroes of $f$ in $D(0, r)$. Show that for ...
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2answers
30 views

Taylor expansion of the series

function f is given by an equation: $$f(x)=\frac{1}{3+x^3}$$ Find the taylor expansion in a point $x_0=0$ and calculate radiu of the convergence. Could you explain how to find taylor expansion of such ...
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3answers
125 views

Prove that $\sum_{n=1}^{\infty}{f\left({1\over n}\right)}$ converges absolutely.

Let $f:\Bbb{R}\to \Bbb{R}$ be continuously twice differentiable around $0$ such that $f(0)=f'(0)=0$. Prove that $\sum_{n=1}^{\infty}{f\left({1\over n}\right)}$ converges absolutely. What I did to ...
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2answers
43 views

Show that if $x_n\rightarrow\ x_0^{-}$ and $y_n\rightarrow\ x_0^{+}$ then $ \lim \limits_{n \to \infty} \frac{f(y_n)-f(x_n)}{y_n-x_n} = f^\prime(x_0)$

Let $f: \Bbb[a,b]\rightarrow\Bbb{R}$ be differentiable in a point $a<x_0<b$. Use Taylor's expansion with the remainder of $f$ around $x_0$ to show that if $ x_n\rightarrow\ x_0^{-}$, and $ ...
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1answer
29 views

Taylor expansion of $\frac{1}{(1-x)^2}$

How do I compute the Taylor expansion of $$\frac{1}{(1-x)^2}$$ around $0$?
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17 views

Approximations of the kind $x<<y$

I have an expression for a force due to charged particle given as $$F=\frac{kQq}{2L}\left(\frac{1}{\sqrt{R^2+(H+L)^2}}-\frac{1}{\sqrt{R^2+(H-L)^2}}\right)$$ where $R$, $L$ and $H$ are distance ...
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1answer
30 views

Prove $\lim_ \limits{x\to x_0}{r_n(x)\over (x-x_0)^n}=0$.

Let $f: I \to \Bbb{R}$ be differentiable $n$ times at $x_0\in I$, and let $r_n(x)$ be the remainder of order $n$ that in the Taylor formula of $f$ around $x_0$. Prove $\lim_ \limits{x\to ...
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1answer
36 views

Radius of Convergence of Power Series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$

What is the radius of the power series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$? Justify your answer. My steps toward a solution I can express $\tanh$ simpler as: \begin{align*} \tanh z ...
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2answers
26 views

What does the Taylor's Inequality mean?

Taylor's Ineqaulity If $|f^{(n+1)}(x)|\leq M$ for $|x-a|\leq d$, then the remainder $R_n(x)$ of the Taylor series satisfies the inequality $$|R_n(x)|\leq ...
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38 views

Differentiable $N$ times, $f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0$, $f^{(N)}(x_0)>0\implies f$ increasing on $x_0$?

Let $I\subseteq \mathbb {R}$ be an open interval and $f:I\rightarrow \mathbb {R}$ is differentiable $N$ times in $x_0\in I$. It's given that: $$f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0, \qquad ...
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0answers
39 views

For which algebras Taylor series and the Fourier series can be generalized?

I'm not a professional mathematician. The question is in the title. But most of all I'd like to know about this for quaternions algebra with non commutative multiplication. I'd like to know about ...
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1answer
27 views

How can I find the convergence radius for this series?

I want to find out the MacLaurin series of this function and find out for which $x$ it equals the original function: $f(x)=\frac{x}{1+3x^2}$ AFAIK I can use this equation: ...
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1answer
17 views

shift taylor series coefficient

Let say I have analytic function $f(z)$ with taylor series $\sum a_nZ^n $ I want to find function $g(z)$ that It's taylor is $\sum a_{n+1} Z^n $ I need that for every $n>1$ : $g ^{(n-1)}(z)$ = ...
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1answer
22 views

Show that a function takes the following form using the definition for the function of an operator

If $f(z)$ is a function with a Taylor series expansion $f(z)=\sum _{ n=0 }^{ \infty }{c_n z^n }$, then we define $f(M)=\sum _{ n=0 }^{ \infty }{c_n M^n }$ First consider ...
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2answers
50 views

How to calculate $\lim \limits_{x \to 0^{+}} (sinx)^{e^{x}-1} $ with Taylor series?

I want to calculate $\lim \limits_{n \to 0^{+}} (\sin x)^{e^{x}-1} $ by using Taylor's Series, and here is what I did so far, and correct me if I'm wrong: $\sin x = x + o(x)$ $e^{x}-1= x + o(x)$ ...
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2answers
34 views

Bounds for $\log(1-x)$

I would like to show the following $$-x-x^2 \le \log(1-x) \le -x, \quad x \in [0,1/2].$$ I know that for $|x|<1$, we have $\log(1-x)=-\left(x+\frac{x^2}{2}+\cdots\right)$. The inequality on ...
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1answer
49 views

How to Taylor expand $(a+b)^n$.

I don't know how to taylor expand $(a + b)^n$ ,can someone send me the proof?