Questions tagged [taylor-expansion]

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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52 views

Why can we just substitute in expressions into Taylor Series?

Say we have the function $(1+x)^{-1/2}$. Using a Taylor Series centered on $x_0=0$, its easy to see that: $$(1+x)^n\approx1-\frac{1}{2}x+\frac{3}{8}x^2+...\mathcal{O}(x^3)$$ In the above, $\...
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3answers
47 views

Taylor series for $x\sin(x^2)$, looking for $a_n$

I had to find the coefficient $a_n \in \mathbb{R}$ so that $\sum\limits_{n=0}^\infty a_nx^n=x\sin(x^2)$,$\forall x \in \mathbb{R}$ Hmm I thought since: $\sin(x)= \sum\limits_{n=0}^\infty(-1)^n\frac{...
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Taylor expansion on a Manifold

Is there a way to define the Taylor expansion of a function $f:\mathcal{M}\rightarrow\mathbb{R}$, where $\mathcal{M}$ is a smooth manifold? I'm looking for a free coordinate definition. I guess it is ...
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23 views

Understanding a derivative to function conversion in a book

I am trying to understand a derivation in a physics book, but I am stuck on a mathematical step. I will reproduce the formulae below in simplified form. However, I can also rewrite the complete ...
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Why Uniformly convexity and taylor's formula implies this in Lawrence C. Evans PDE?

In the proof of Lemma 4(Semiconcavity again) in section 3.3.3 of the book "Partial differential equations" written by Lawrence C. Evans, "We note first using Taylor's formula that (35) implies (36) ...
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68 views

Evaluate: $\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$

The questions asks to evaluate: $$\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$$ The answer provided is $-\frac{1}{4}$, but I don't know how to solve it. I am not sure how to ...
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2answers
40 views

Evaluate $\sum_{n=0}^{\infty} \frac{{\left(\left(n+1\right)\ln{2}\right)}^n}{2^n n!}$

Evaluate: $$\sum_{n=0}^{\infty} \frac{{\left(\left(n+1\right)\ln{2}\right)}^n}{2^n n!}$$ I am not sure where to start. The ${\left(n+1\right)}^n$ term is obnoxious as I can't split the fraction. ...
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4answers
43 views

Using the Maclaurin series for $\frac{1}{1-x}$ to find $\frac{x}{1+x^2}$

Suppose I know the Maclaurin series for $$\frac{1}{1-x}=1+x+x^2+x^3+...= \sum_{n=0}^{\infty}x^n \tag{1}$$ then I can find the Maclaurin series for $\frac{1}{(1-x)^2}$ by the substitution $x\to x(2-x)$,...
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Find the Taylor-series for $f$ in $0$

I'm dealing with the following problem. Let $a\in\mathbb{R}\setminus\{0\}$ and define $f:{\mathbb{R}\setminus\{a\}}$ with $$f(x)=\frac{1}{a-x}, \hspace{20pt} x\in\mathbb{R}\setminus\{a\}$$ Find the ...
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Finding Maclaurin series f(x)

Can anyone please help me with finding Maclaurin series for this $$f(x) = x^3 \tan^{-1}(2x); \quad |x|<\frac12$$ https://i.stack.imgur.com/bUhxk.jpg
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Taylor expansion upper bound

Let $x > 0$ and let $c < 1$ be some constant. I am wondering: can I find an upper bound on $x$ such that $$ e^{-x} + (1 - c)x < 1$$ which is a function of $c$? For instance, something like $$...
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Analyticity of Integral function

I have recently read most of Conway's book on Complex Analysis and now I'm trying to know in which conditions $f: D \to \mathbb{C}$ is complex analytic (I'm mixing two concepts from 2 of his chapters):...
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Find $a \in \mathbb{R}$, so that $|h(x)-T_2(h,x,\sqrt{\pi})|\le a|x-\sqrt{\pi}|^3$, $\,\,\forall x \in [0,2]$

Find $a \in \mathbb{R}$, so that $|h(x)-T_2(h,x,\sqrt{\pi})|\le a|x-\sqrt{\pi}|^3$, $\,\,\forall x \in [0,2]$ $h:\mathbb{R} \longrightarrow \mathbb{R}:x \mapsto x\sin{(x^2)}$ My attempt: $|h(x)-T_2(...
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1answer
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Different approximations of this function

Was told to post here. However, I have heard about this site, as well, but I am hesitant on posting on the internet, hence I made an account. Anyway, my question is: How can I approximate the sin(2) ≈...
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1answer
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Taylor expansion of $e^z$ - some doubts on an approximation regarding terms from the third-order one onward

I quote Jacod-Protter herebelow. Let $\left(X_n\right)_{n\geq1}$ be a sequence of Poisson random variables with parameter $\lambda_n=n$. ($\ldots$) We have \begin{equation*} \begin{split} E\...
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Expansion of trace of matrix

Assume a symmetric matrix $W_{n \times n}$ with parameter matrix $P_{n \times n}$. $P_{n \times n}$ is estimated by $\hat P_{n \times n}$ and plug it in $W$, write the estimated matrix as $\hat W_{n \...
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1answer
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Taylor series expansion of function $f$ order $n$ at $x_0$

Hello please help me about this one $$\frac { 1 + e^{ - 1 / x^{ 2 } } } { 2 + x },$$ order $n=5$ at $x_0=+\infty$. Because this lesson I just learn it so help to show me. Thank in advance! Since : $$...
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44 views

Taylor serie for even function. Proof [duplicate]

We let $f:\mathbb{R}\to\mathbb{R}$ be infinitely often differentiable function and we let the Taylor series be: $$\displaystyle\sum_{n=0}^{\infty}\left(\left(\frac{f^{n}(0)}{n!}\right)x^n\right) $$Let ...
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How to taylor expand a tensor?

Question Is there some nice way to Taylor expand a tensor in a coordinate invariant way? (I expect some christoffel symbols would be involved) After searching on the internet I found this however, ...
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3answers
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Closed for solution for $\sum_{k = 0}^{n} Q^{k} ( 1 - Q) ^ {k}$

I know the binomial expansion formula: $$ (1 + x)^n = \sum_{k = 0}^{n} {n \choose k}x^k $$ However, I am trying to find (if there is any) a closed-form solution for the following equation. $$ \sum_{...
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'Taylor Expansion' of Integral - Asymptotic expansion - Exponential function

I need to evaluate the following integral in the limit $\kappa \ll 1$ $$\int_0^\infty exp(-\kappa t) f(t)\, dt,$$ where $$f(x) = (1+x)(1-2x)\frac{u(x) \ln(u(x))}{u(x)^2 - 1},$$ $$u(x) = \frac{\sqrt{1+...
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Taylor series at $a=1$

I have to find Taylor series at $a=1$ for $ f(x)=\begin{cases} \frac{e^{x}-e}{x-1},\quad &\text{if } x\ne1\\ e,\quad &\text{if } x=1\\ \end{cases} $ I ...
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1answer
37 views

Convergent on edge. 1/a

For $a \in \mathbb{R}\setminus\{0\}$ we have function $f$:$\mathbb{R}\setminus\{a\}\rightarrow \mathbb{R}$: $$f(x)=\frac{1}{a-x}$$ for $x\in \mathbb{R}\setminus\{a\}$. Then I have to find the Taylor ...
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How to find the Taylor series for this function? [closed]

$f(x)=\frac{\cos{x}}{2x-\pi},$ when $x\neq\frac{\pi}{2}$ and $f(x)=-1/2,$ when $x=\frac{\pi}{2}$ $x_0=\frac{\pi}{2}$ I am very confused by the condition set by the exercise. I don't know if I should ...
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55 views

Taylor series for $f(x)= \sqrt[5]{3+2x^3}$ at $a=0$

I have to find Taylor series representation $\sum_{n=0}^\infty a_nx^n$ for the function $f(x)=\sqrt[5]{3+2x^3}$ where $a_n=\frac{f^{n}(0)}{n!}$. The series itself is easy to calculate $(f(x)=\sqrt[5]...
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Series with double factorial

Prove that $$\frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} = \frac{1}{2} + \sum\limits_{k=1}^{\infty}\frac{(4k-1)!!}{(4k+2)!!}x^{2k} , \forall x \in (-1, 1)$$ where $x!!$ means double factorial in this case....
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Find the radius of convergence for infinite series. [duplicate]

I have just shown via the Taylor expansion for $\sin(\frac{1}{n})$ that the series $$ \sum_{n=1}^{\infty}\left(\frac{1}{n} - \sin\left(\frac{1}{n}\right)\right) $$ is in fact convergent and now I'm ...
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35 views

Trying to make a list of important Taylor series

I am trying to come up with a list of series and Taylor series I should probably know before I take my qualifying exam in august. Here is what I got, please let me know if one of them is wrong or if ...
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1answer
56 views

Taylor series of $\ln(1+x)$ when $x=1$

I know what the Taylor Series of $\ln(1+x)$ is, but, I don't know why it's true also for $x=1$. If I decide to use the method of integration starting from $\frac{1}{1+x} \ =\ \sum ( -1)^{n} x^{n}$ ...
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1answer
38 views

Can someone recommend for me a text-book about applications of Taylor series?

I have text-books about Taylor series but they do not mention Taylor series applications. Can anyone please suggest a few references to learn the same?
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If even function then … [duplicate]

We let {$a_n$}$_{n\in N}$ be $a_n$=$\frac{f^{n}(0)}{n!}$. I have to show that if $f$ is an even function so is $a_{2n-1}$$=0$ for all n$\in$N. How can I show it? By induction maybe? Can anyone give a ...
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1answer
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Find the solutions to the $w''-z^2w=3z^2-z^4$ as Taylor series where $w(0)=0$ and $w'(0)=1$

We need to find the solutions of the $w''-z^2w=3z^2-z^4$ where $w(0)=0;w'(0)=1$ I wrote down the series that we can use to find the answer ($w$ as Taylor series): $w=\sum_{n=0}^\infty C_nz^...
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1answer
38 views

Find Taylor serie and radius of convergence

For a$\in$R/{0} we have function $f$:$R$/{a}$->$R: $$f(x)=\frac{1}{a-x}$$ for x$\in$$R$/{a}. Then I have to find the Taylor serie in 0. I think it's maybe: $\displaystyle\sum_{n=0}^{\infty}(\frac{n!...
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1answer
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Production expresion for Taylor series expansión.

i need help, I am doing the Taylor series development around the origin of: $\sqrt{z+i}$, i have considerated $\sqrt{z+i}=\sqrt{i} \sqrt{1+\frac{z}{i}}$ and make the variable change $x= \frac{z}{i}$ ...
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1answer
102 views

Calculating $\lim_{x\to \infty} (x+1) \cos \left(2\cos ^{-1}(\frac{x}{x + 1}) (a - \frac{1}{2})\right) - x$ using cosine expansions

Question is: $$\lim_{x\to \infty} (x+1) \cos \left(2\cos ^{-1}(\frac{x}{x + 1}) (a - \frac{1}{2})\right) - x$$ The answer is $4x-4x^2$ but I'm not sure how to get there. I have to use expansions of ...
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Evaluating erf(x) using Taylor's series

I tried to evaluate error function using Taylor series by using its definition $$ erf(z) = \frac{2}{\sqrt{\pi}}\int_0^ze^{-t^2}dt$$ I've used Taylor expansion to evaluate this integration and i got ...
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Prove that $\exp(x+y) = \exp(x)\exp(y)$. [duplicate]

For every real number $x$, we define the exponential function $\exp(x)$ to be the real number \begin{align*} \exp(x) = \sum_{n=0}^{\infty}\frac{x^{n}}{n!} \end{align*} Prove that $\exp(x+y) = \exp(x)\...
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Taylor series $\sqrt{z+i}$

could you please help me to develop the following serial function of taylor, i can't find the pattern of the nth derivatives, i would appreciate it very much: $\sqrt{z+i}$ I've dealt with common ...
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1answer
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Show that $|\sin(0.1) - 0.1| \leq 0.001$ with the lagrange remainder

Show that $|\sin(0.1) - 0.1| \leq 0.001$ I know that's a basic exercise on taylor polynomial but I have made a mistake somewhere that I don't find out. Anyway, here's my attempt : Because the ...
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0answers
15 views

Scaling as a repeated shift

I was wondering if the following derivation of the scale operator starting from the shift operator is good. Everybody knows that an operator $T_a$ acting on a function $f$ as $$ T_af(x)=f(x+a) $$ can ...
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1answer
27 views

$f^{(k)}(x)/|x|^{n-k} \to 0$ as $x \to 0$ when $f \in C^{n}(\mathbb{R})$ and $f^{(k)}(0)=0$ for all $0 \leq k \leq n$

I was thinking about the exercise mentioned in the title: Let $f \in C^{n}(\mathbb{R})$ for some $n \geq 0$ and further suppose that $f^{k}(0)=0$ for each $0 \leq k \leq n$ then show that $f^{(k)}(...
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How do we prove that $\int_0^1 (x\ln x)^{-1+n}\,dx = -\left( -\frac{1}{n} \right)^n \Gamma(n)$

I need this solution to prove that $$\int_0^1 (x\ln x)^{-1+n}\,dx = -\left( -\frac{1}{n} \right)^n \Gamma(n)$$ Thank you!
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1answer
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$f(Z)=\log((1+z)/(1-z))$,$z\in \mathbb{C}$ it can expand a series of taylor

I have to prove that $f(Z)=\log((1+z)/(1-z))$,$z\in \mathbb{C}$ it can expand a series of taylor like $f(z)=2\sum_{n=0}^{\infty} ( x^{2n+1})/(2n+1)$ for all $z\in D(0,1)$ my idea is to drift ...
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1answer
47 views

Knowing the expansion of a function, how can we find its expansion using the inverse of x?

If we have a function like: $$\text{f[x$\_$]:=}\sum _{i=0}^{\infty } a_ix^i$$ where we can find / know the $a_i$ coefficients, but not really for which function it will converge. How can we find $f[...
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1answer
44 views

Find $\lim \limits_{x \to 2} {sin(x)cos(x)e^{cos(x)}\over ln(x)}$ [closed]

The limit can be found if I use a calculator but how do I find it without using one? I tried to use the Taylor' series and this is what I have so far but it doesn't look right: $${(x-{x^3 \over 3} + ...
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2answers
59 views

Find limit if it exists and explain why if it does not: $\lim_{x\to 0} \cos(1/x) e^x$

I started with the Taylor expansions of $\cos(1/x)$ and $e^x$ i.e $\cos(1/x) : [ 1 - \frac{1}{2x^2} + o(\frac{1}{x^4}) ]$ and $e^x : ( 1 + x + \frac{x^2}{2} + o(x^3) )$ I realise $e^x$ goes to $...
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0answers
12 views

Lower bound for log normalizer of a Dirichlet distribution

The log-normalizer of a Dirichlet distribution is defined as: $$ \log Z = \log \Gamma (\sum_k u_k) - \sum_k \log \Gamma (u_k), $$ where $\textbf{u} \in R^K $. In my case, $\textbf{u}$ is a log-...
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2answers
50 views

Taylor Series of $f(x) =\frac{1}{x^2}$

I've found the Taylor Series for $f(x)=\frac{1}{x^2}$ centered at $a=-1$. $f(-1)=1$, $f'(-1)=2$, $f"(-1)=6$, $f'''(-1)=24$, $f^4(-1)=120$ I used this formula to get each the first coefficients of ...
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0answers
40 views

A holomorhic function $f $ in the unit disc such that $\lim_{z\rightarrow 1}f(z)$ does not exist

Let $f$ be a holomorphic function in the open unit disc such that $\lim_{z\rightarrow 1}f(z)$ does not exist. Let $\sum_{n=0}^{\infty}a_nz^n$ be the Taylor sereis expansion of $f$ about $z=0$ and $R$ ...
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1answer
22 views

What is the power series of $1/(D-h(x))$ if $h(x)\ll D$?

I have a problem, which I do not conceptually understand. I need to approximate an arbitrary function $$\frac{1}{D-h(x)}$$ where $h(x)$ is arbitrary, $h(x)\ll D$ and $D$ is a constant. Friends say ...

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