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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is it true that $ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx - \frac{1}{1-\epsilon}x $?

Target is to approximate $\frac{1}{\epsilon}\ln (1- \epsilon x) $ ($\epsilon, x \in (0,1) $). Here is one using $\ln (1+y) \approx y $: $$ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx -x $$ I ...
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1answer
49 views

Evaluate thus limit using series: $\lim_{x\to0} (\sin x-\tan x)/x^3$

Evaluate thus limit using series: $$\lim_{x\to0} \frac{\sin x-\tan x}{x^3}$$ I know the value of this limit is -1/2, and I also know the series expansion for $\sin x$ is $$x - \frac{x^3}{3!} + \frac{...
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1answer
30 views

Derivatives of characteristic function

Let $\phi$ be the characteristic function for random variable $X$. I know that if $E [|X|] < \infty$, then dominated convergence implies existence of the first derivative, and in particular, $\phi'(...
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1answer
81 views

Approximating polynomial of higher degree

Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values of ...
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2answers
45 views

Taylor Series and Differentiation with Sigma notation $f(x) = \frac{x}{(2-3x)^2}$

Use Term By Term Differentiation to Find the Taylor Series about $x$=3 for Give The Open Interval of Convergence and express as sigma notation $\sum A_n(x-3)^n$ $f(x) = \frac{x}{(2-3x)^2}$ So I ...
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1answer
47 views

Value of the limit without (or with, but giving rigorous arguments) using the Taylor expansion of sin

I'm trying to evaluate the limit as $N\to \infty.$ $$\frac{ \left(\dfrac{\sin \frac{1}{N}} {\frac{1}{N}}\right)^{N} -1 }{\frac{1}{N}}.$$ Note first that, using L'Hôpital, one can easily show that ...
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3answers
75 views

Expand in Taylor series $\frac{1}{1-\sin{x}}$

Expand in Taylor series $\frac{1}{1-\sin{x}}$ I have an idea that $\frac{1}{1-\sin{x}} = 1 + \sin {x} + \sin^2 {x} + \sin^3 {x} + \dots$ But I don't know what to do next. Every sine expands in ...
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1answer
5k views

Series expansion of square root function

Could I please know how to expand: $$\sqrt{4-3x^2}$$ I simplified it to $\sqrt{1-\frac34x^2}$ but the question is if I let $x = x^2$, what will the coefficient of, say $x^5$ or $x^{10}$ be in the ...
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44 views

Find smallest $k$ such that the given trigonometric functions are $O(x^k)$

I feel like I do not quite grasp the concept of Big O Notation. From my understanding, if $f(x) = O(g(x))$ then $f(x)$ is at most $g(x)$ multiplied by some constant C, which makes decent sense to me. ...
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1answer
38 views

Logarithmic expansion with cosines

I found the following expansion in this paper: $$\log\frac{|\boldsymbol{r}-\boldsymbol{r'}|}{L}=\log\frac{r_>}{L}-\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{r_<}{r_>}\right)^n\cos[n(\phi-\phi'...
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1answer
35 views

Does point's neighborhood have no local extremum?

I have polynomial of some limited degree: $f(x,y) = a_1 + a_2x + a_3y + a_4xy + a_5x^2 + a_6y^2 + a_7x^2y + \ldots$ There is a point $p_0=(x_0,y_0)$, which is NOT a local extreme NOR inflection ...
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2answers
61 views

Find a bound for approximating Taylor Series

I'm struggling to figure out how to find a bound on my error for this problem: Let T_{6}(x) be the Taylor polynomial of degree 6 based at a = 0 for the function f(x)=\cos(x). Suppose you approximate ...
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0answers
30 views

Order Taylor series prediction

It is easy to add the Taylor expansion, less easy to multiply and even less easy to compose. That said, the main problem lies not in the calculation itself in the prediction orders that need to be ...
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1answer
33 views

Accuracy of Runge-Kutta compared to Taylor expansion

Let's say I have an ODE of the form : $y'(x) = f(x,y(x))$. I've been told that using the Runge-Kutta method for solving this ODE is equivalent to using Taylor expansion if $f(x,y(x))$ is linear in $y$ ...
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13 views

Sommerfield Expansion Taylor Expansion

Ugh... I can't figure this out and I DO NOT understand why. So this has to do with the Sommerfield expansion of the Fermi function (wiki) (another reference) The issue I'm having is we are supposed ...
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1answer
75 views

Let $s(n,k)$ denote the signless Stirling numbers of the first kind. Prove that…

Let $s(n,k)$ denote the signless Stirling numbers of the first kind. Prove that: $$s(n,2) = (n-1)!(1 + \frac{1}{2} + \frac{1}{3} +...+ \frac{1}{n-1})$$ -I haven't dealt with Taylor series expansion ...
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1answer
40 views

Second order differential

Suppose I have a function $f = f(x,y,z)$. Then, the first order differential, or the linear approximation is $$ df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial ...
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0answers
31 views

Asymptotic behavior of the function $e^{- \lambda t^2}$ when $\lambda$ is small

I wish to prove that when $\lambda$ is taken to be very small $$ \left| e^{- \lambda t^2} - \sum_{n=0}^N \frac{(- \lambda t^2)^n}{n!} \right| = O(e^{-\frac{a}{\lambda}})$$ for some constant $a \in \...
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2answers
67 views

prove $(1+\ldots+o(x^{n-1}))^4=(1+\ldots+o(x^{n-1}))$

I would like to prove that : $$(1+\ldots+o(x^{n-1}))^4=(1+\ldots+o(x^{n-1}))$$ i took that from the picture below My Proof: note that $$(1+x)^a = 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-...
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1answer
44 views

Differentials squared - Divergence in general orthogonal curvilinear coordinates.

I was reading this document on how to get some common operators when dealing with general orthogonal curvilinear coordinates. I am interested in particular in equation (12). It basically defines three ...
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2answers
30 views

$\exists C>0$ such that $\frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2-Cx}$?

It is well-known and easy to check that for any real $x$ it holds $$ \frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2}. $$ [To show this, it is sufficient to write explicitly their Taylor series ...
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1answer
19 views

Two dimensional taylor expansion of arbitrary function

Consider the function dependent on the variables $N_t$ and $N_{t-1}$. Call the function $f$ so $f = f(N_t, N_{t-1})$. Now suppose we could write $N_t = N^*+n_t$ where $N^*$ is constant, and $n_t$ ...
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1answer
21 views

Show a function behaves as a harmonic oscillator

We have a function $V(x)$ (potential energy) with $x$ being some variable. This function has a minimum at a certain $x_0$. We assume that $V(x)$ is an analytic real function of $x$ around $x_0$. ...
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2answers
37 views
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1answer
36 views

Use Taylor Series to arrive at the expression for the forward approximation for a derivative [duplicate]

Use Taylor Series to arrive at the expression for the forward approximation for a derivative. $$f'(x)\approx \frac1h\left(-\frac32f(x)+2f(x+h)-\frac12f(x+2h)\right)$$ I'm not sure how to even go ...
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1answer
98 views

Integral of $\int x^{-x} dx$

Question: $\int x^{-x} dx =$ ? Hint: $$ e^{x\ln \frac{1}{x}} = \sum_{n=0}^\infty \frac{x^n}{n!} \left(\ln\left(\frac{1}{x}\right)\right)^n$$ I figure since $\int x^{-x} dx = \int e^{x\ln \frac{1}{x}...
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2answers
80 views

Prove that $\int _ {a}^{b}x^n f(x)dx=0$ implies $f=0$ by using the Taylor theorem

Let $f$ be a smooth function on a closed interval $[a,b]$. Assume that there is $M>0$ such that $\left|f^{(n)}\right|<M$ for all $n\geq1$ and all $x\in[a,b]$. Prove that if $$\int _ {a}^{b}x^n ...
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0answers
14 views

How can I show that these 2 Taylor series expansions are equivalent (in 2 dimensions)?

I've been given the following questions: For part $(i)$ I've found that the expansion is given by $f(x+h, y+k)= 1+2h+k+3h^{2}+2hk+\frac{k^{2}}{2}$. However, for part $(ii)$ I've found that $g_{1}(...
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1answer
35 views

Why is there a transpose in multivariable Taylor formula? [closed]

Regarding Taylor polynomial of 2nd order, $$ f(x)=f(x_1)+\operatorname{grad}(f(x_1))^T (x - x_1)+\frac12(x - x_1)^T \operatorname{grad}^2f(x_1) (x - x_1) $$ I have the following question: Why is ...
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0answers
38 views

Smoothness of $x\mapsto \frac{1}{1-x}$

I'm reading lecture about taylor expansion but i wonder in following example why he took $x\longmapsto \dfrac{1}{1-x}$ as function in $]-\infty,1[$ of class $\mathcal{C}^{n+1}$ and not in $\mathbb{R}...
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3answers
32 views

$\frac{1}{(1-x)^{2}}=\sum_{k = 0}^{n}(k + 1)x^k+o(x^{n}).$

I would like to show that Taylor expansion of $\dfrac{1}{(1-x)^{2}} $ around $0$ is : $$\dfrac{1}{(1-x)^{2}}=\sum_{k = 0}^{n}(k + 1)x^k+o(x^{n}).$$ My Proof: note that $$\dfrac{1}{1-x}\...
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2answers
24 views

Question about the Taylor theorem and different ways of expressing it

In my lecture notes, it says that the Taylor theorem in one dimension is given by $$f(x+ \delta x) \simeq f(x)+\delta x f_{x} + \frac{1}{2!} (\delta x)^2 f_{xx}+ \dots$$ Conversely, when I was back ...
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1answer
144 views

Taylor series expansion of $e^{-\lambda\tau}$

Taylor series expansion of $e^{-\lambda\tau}$ yields: $e^{-\lambda\tau}=1-\lambda\tau+o(\tau)$ and Taylor expansion of $\lambda\tau e^{-\lambda\tau}$ equals: $\lambda\tau e^{-\lambda\tau}=\lambda\...
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1answer
33 views

Real Analysis: Bounds for derivatives using Taylor's Theorem

Suppose that $f''$ exists on [0,1] and that $f(0)=0=f(1)$. Suppose also that $|f''(x)|\leq K$ for $x\in(0,1)$. Prove that $|f'(1/2)|\leq K/4$ and that $|f'(x)|\leq k/2$ for $x\in(0,1)$. I'm trying to ...
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0answers
31 views

Finding a Maclaurin series

I have a question here; suppose $f(x)= x^2\sin(x^3)$ By using the Maclaurin series for sine, find the Maclaurin series for $f$ I understand how to obtain the Maclaurin series for $f$ using the ...
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2answers
59 views

Analytic functions equal to all orders in a point are equal on the open interval

Let $A\subset \mathbb{R}$ be open. To make everything clear, my definition of analytic function here is: A function $\psi : A\to \mathbb{R}$ of class $C^\infty$ is said to be analytic if for each $...
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77 views
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1answer
71 views

Asymptotics and little-o notation

I always have issues dealing with asymptotic notation... I am trying to verify the following step: $$\left(1-\frac{t^2}{2n} + o(1/n)\right)^n \to e^{-t^2/2}.$$ To change this into $(1-t^2/(2n))^...
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1answer
40 views

How to show the existence of an entire function

I have been working on this problem for quite sometime. For part (i), I obtained the Taylor series for $4\sin(z) - \sin(4z)$. At $z = -\pi$, the Taylor series is: $4\sum_{n=0}^{n} \frac{(z + \pi)^{...
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2answers
49 views

Compact Form of the Taylor Series

Determine the Taylor Series $\frac{1}{\sqrt{1-x}}$ at $x=0$ I ended up with this: $1 + \frac{1}{2}x+\frac{3}{4}x^2\frac{1}{2!}+\frac{15}{8}x^3\frac{1}{3!}+\frac{105}{16}x^4\frac{1}{4!}$ I am ...
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1answer
90 views

Limitations of fractional derivative approximation with Taylor series

I was playing around with the concept of fraction derivatives, and came across some base functions for which it is defined, namely power and exponential functions $$ \left(\frac{d}{dt}\right)^\alpha ...
2
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2answers
98 views

taylor expansion of $\sinh(x)$

I would like to find taylor expansion of $sh(x)$ My thoughts indeed, note that : $\sinh(x)=\dfrac{e^{x}-e^{-x}}{2}$ then \begin{align} \sinh(x)&=\frac{e^x-e^{-x}}{2} \\ &=\frac{1}{2}\...
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1answer
61 views

Different convergence radius for different power series of the same function.

i was playing around with $$\frac{1}{x^2+x+1}$$ I got 3 different series's: $$\sum_{n=1}^\infty (\frac{x}{(x+1)^2})^n*(\frac{1}{x}) $$ which converges when $|\frac{x}{(x+1)^2}|<1$ the second ...
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2answers
52 views

To show that the limit of the sequence $\sum\limits_{k=1}^n \frac{n}{n^2+k^2}$ is $\frac{\pi}{4}$

Show that $$\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{n}{n^2+k^2} = \frac{\pi}{4}.$$ I am familiar with Taylor series and Fourier series of the standard functions. I tried to compare with those ...
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2answers
39 views

Taylor series for df

So I understand if I have f(x) under a taylor expansion I can write the terms up to order 2 terms as: f(x)= f(a) + f'(a)(x-a) + [f''(a)*(x-a)^2]/2! +... so I would imagine df(x)/dx = f'(a) + [f''(a)...
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0answers
26 views

$\ln(1+x)\underset{x\to 0}{=} \sum_{k=1}^{n}(-1)^{k+1}\frac{x^{k}}{k}+o(x^{n})$

I would like to see the setps behind that implication $$ \frac{1}{1+x}\underset{x\to 0}{=} \sum\limits_{k=0}^{n-1}(-1)^{k}x^{k}+o(x^{n-1}) \implies \ln(1+x)\underset{x\to 0}{=} \sum\limits_{k=1}^{n}(...
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1answer
49 views

Leading order Taylor Series Represention of the following function

I am given with this function $$f=\frac{1}{\sqrt{1+af_1(x)+bf_2(x)}},$$ where $$f_1=(1+x^2)^\nu,$$ and $$f_2=x^2(1+x^2)^{\nu-1},$$ where $\nu$ is a rational constant. I would want my $f$ to be of the ...
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2answers
49 views

Taylor expansion for vectors

$$F(x,y)= (x_2-x_1^2) (x_2-2x_1^2)= 2 x_1^4+x_2^2-3x_1^2x_2$$ Where $x^*=[x_1 \ \ x_2]' = [0 \ \ 0]'$ I want to show Taylor expansion of the function for third degree. What I did is that; ...
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1answer
23 views

Reducing terms in the series expansion of a function of two variables

I have a function $f(x, y)$. This function is such that \begin{align} f(0, y)=a\\ f(x, 0)=a, \end{align} where $a$ is a constant. From this, a particular mathematician concludes: Thus if we ...
4
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1answer
85 views

On the binomial series $(1+\frac{1}{8n})^{1/2}$, where $n$ is an even perfect number

Since $\sqrt{1+8n}=\sqrt{8n}\sqrt{1+\frac{1}{8n}}$, and $\frac{1}{8n}<1$ when $n>1$ is an integer, then we can express the real number $\sqrt{1+\frac{1}{8n}}$ by its binomial series. This series ...