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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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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48 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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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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4answers
48 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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1answer
24 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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2answers
48 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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22 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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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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26 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
29 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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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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119 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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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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16 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
29 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
31 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
24 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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2answers
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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33 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
15 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
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
21 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
30 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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2answers
49 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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1answer
48 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?
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67 views

Fundamental proof of Taylor's theorem using little-o notations

Is there a fundamental proof of Taylor's theorem using little-o notation? I assume $f:E\rightarrow F$ as a mapping between Banach spaces and write $(h^i)$ for $(h,\ldots,h)$ ($i$ times iterated). ...
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1answer
24 views

Prove Taylor expansion converges to $f(x)$ for all $x\in I$ fulfilling $|x-x_0|<{1\over M}$.

Is a single piece data incorrect? How can I know? Let $f\in C^{\infty}(I)$, and let $C,M >0$ such that $|f^{(n)}(x)|\le CM^n n!, \forall x\in I, n\in \Bbb{N}$. Prove that for all $x_0\in I$, ...
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Taylor Series, Remainders, and Uniform Convergence

This is a question about Taylor Series in general, but I want to point to a specific example: $$e^x = \sum_{n = 0}^\infty \frac{x^n}{n!} = \sum_{n=0}^\infty f_n, \hspace{.25in} x \in \mathbb{R}$$ ...
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1answer
18 views

Expansion of cumulant transform

Verify the following expansion for a cumulant generating function of a random variable $X$. \begin{align} \kappa(t) & = \mu t + \frac{1}{2}\sigma^2t^2+\frac{1}{6}\rho_3\sigma^3t^3 + ...
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1answer
26 views

Integer Series Expansion

For any two $p,q \in \mathbb{Z}$ and $n \in \mathbb{Z}^+$, can one prove that $a_n = \frac{p(-p)^n - q(p-2q)^n}{(p-q)}$ is an integer with recursion relation $a_0 = 1,$ $a_n = ...
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Can we always exploit Taylor's approximation in order to avoid using l'Hopital rule?

We are given two functions, namely: $f, g: U^\circ\subset \mathbb{R}\to \mathbb{R}$ where $0 \in \overline{U}$. Suppose you want to compute the limit $$\lim_{x \to 0} \frac{f(x)}{g(x)}$$ and that ...
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1answer
50 views

Taylor series for $\frac{1}{(1+x)^t}$

I'm having some trouble finding the Taylor series for the following function at zero (Maclaurin series). \begin{equation} \frac{1}{(1+x)^t} \end{equation} Where $t$ is a constant that is greater than ...
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1answer
47 views

$f(x)={1\over (x-1)(x+2)}$ Taylor series.

Find Taylor series around $x_0=0$ for: $$f(x)={1\over (x-1)(x+2)}=(\text{By a hint\by simple algebra}){1\over 3}\left[{1\over x-1}-{1\over x+2}\right]$$. Check where the series converges to the ...
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0answers
19 views

Minimising error in iterative method

Consider the method $$y_{i+1}=y_i+h\left(\frac{1}{3}f(y_i)+af(y_i+hbf(y_i))\right)$$ as applied to $y'=f(y)$. How should $a$ and $b$ be chosen to minimise the local error? I thought maybe the best ...
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1answer
28 views

Quadratic Formula expressed with Taylor's Theorem

I am having trouble solving the problem below. I think I understand the first part by just doing a taylor expansion of $f(\delta - a)$ where $a=0$ and the function equals $\sqrt{1-\delta}$. But I do ...
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1answer
26 views

Taylor series expansion of multiple terms

I'm having trouble getting my head around the expansion of: $$f(y+bhf(y))$$ I want to expand a function within a function. I have been given that ...
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2answers
34 views

Finding Taylor polynomial for fractional function

I should find Taylor polynomial of a degree $n \in \mathbb{N}$ for function $f(x) =\frac{x}{9+x^2}$ at the point of 0. For which $x \in \mathbb{R}$ this polynomial converges to the given function $f$ ...
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Technical Worries with Lagrange Form of Remainder

Here is a typical problem in a beginning class in numerical analysis. (My own words) Let $f: \mathbb{R} \to \mathbb{R}$ be a $C^2$ function on $[x_1,x_2]$. Use Taylor's Theorem to find the local ...
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Applying the Taylor Series to contact of two curves

I am introducing myself to differential calculus and am trying to understand the application of the Taylor series to the contact of two curves. In the book, it says: Let $y=\phi(x)$ and $y=\psi(x)$ ...
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1answer
61 views

A question on smooth functions and the Taylor series

I've been reading Wald's general relativity book and at the end of chapter 2 (question 2) he asks the reader to prove the following expression (multivariable Taylor series) by induction and I'm unsure ...
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1answer
168 views

Branch points of the Lambert W function

Let $W_{k}(z)$ be the kth branch of the Lambert W function. My question pertains to the branch point that $W_{0}(z)$ shares with $W_{-1}(z)$, and $W_{1}(z)$ at $z = - \frac{1}{e}$. By the inverse ...
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1answer
31 views

Expand the function as a taylor series

Consider a random variable $A$ whose expectation value is: $$= \sum\limits_{n= 1}^N {a^m_n}p $$ Expand $f(A)$ as a taylor series that is as $\sum_{j=1}^\infty {c_j}A^j$ for some ...
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2answers
61 views

Find series expansion

I have problem with expanding at $x=0$ function $f(x)=\ln{\arctan(x)}$ I've seen at wolfram and it's $\displaystyle\ln{x}-\frac{x^2}{3}+O(x^4)$ but I don't know how to obtain it via Taylor series I ...
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30 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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1answer
46 views

Series expansion of $\exp(-x)$ using powers of $\frac{1}{1+x}$?

I want a power series expansion of $e^{-x}$, but since powers of x blow up as x→∞ and powers of $\frac{1}{x}$ blow up as x→0, I was wondering if a series expansion of $e^{-x}$ using powers of ...
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1answer
32 views

How to develop in Taylor series

I need to develop around $\pi$ $f(x)=sin(\frac{x}{2})$, and to determinate the convergence interval and the radius of convergence of the serie. I've started by finding all the derivate $$f(x) = ...
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2answers
34 views

Approximating the value of $\frac{1}{\sqrt{1.1}}$ using Linear approximation of $\frac{1}{\sqrt{1+x}}$.

How do I calculate approximately the value of $\frac{1}{\sqrt{1.1}}$ with Linear approximation of the function $\frac{1}{\sqrt{1+x}}$ around point $0$. And here is a follow-up question: Show that the ...
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3answers
34 views

Calculating using Taylor's series with Remainder: $ \lim \limits_{x \to 1} \frac{\ln x}{x^2+x-2} $

How to Calculate with Taylor's series with Remainder: $$ \lim \limits_{x \to 1} \frac{\ln x}{x^2+x-2} $$ without using L'Hopital's Rule? Here is what I reached: $$\lim \limits_{x \to 1} \frac{(x-1) ...
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1answer
21 views

Taylor series of $r:x \mapsto \begin{cases} e^{-{1\over x}}, & \text{if $x>0$} \\ 0, & \text{if $x \le 0$}\end{cases}$ at $0$

Prove the following lemma: The function $$r:x \mapsto \begin{cases} e^{-{1\over x}}, & \text{if $x>0$} \\ 0, & \text{if $x \le 0$} \end{cases}$$ is $C^{\infty}$ (and x=0),that has ...