# 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.

6,347 questions
Filter by
Sorted by
Tagged with
52 views

27 views

### 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 ...
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 ...
13 views

### 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) ...
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 ...
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. ...
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 $x\to x(2-x)$,...
25 views

### 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 ...
23 views

### 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
23 views

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 ...
23 views

### 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, ...
31 views

27 views

### 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 ...
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 ...
31 views

### 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 ...
55 views

28 views

### 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 $x= \frac{z}{i}$ ...
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 ...
55 views

### 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 ...
39 views

33 views

### 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!
### $f(Z)=\log((1+z)/(1-z))$,$z\in \mathbb{C}$ 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 ...