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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Methods for bounding the remainder of taylor expansion for $e^x$

When I was reviewing taylor series, I ran into trouble on the following problem. Problem: Write down a Taylor polynomial to compute $e^x$ within $10^{-3}$ of error on the interval $[-1,2]$. ...
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1answer
34 views

Is there any known way to sum a subserie (square indices) of geometric series?

I was interested in the following sum. Although im not sure there exist any known way to sum this...it seems rather difficult. Can we sum for $0<r<1$ $$\sum_{k=0}^{\infty}r^{k^2}= 1+r+r^4+r^{9}+...
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1answer
27 views

Taylor expand $\ln(x) - \ln(1-y)$ around$(\ln(x'),\ln(y'))$

Can I taylor expand $$\ln(x) - \ln(1-y)$$ around $(\ln(x),\ln(y'))$ such that I get $$ \ln(x') - \ln(1-y') + \frac{\partial (\ln(x) -\ln(1-y))}{\partial \ln(x')} (\ln(x) - \ln(x')) + \frac{\partial (\...
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1answer
24 views

Taylor Maclaurin series

Can someone explain to me how this equals? I'm taking a calculus III course at the moment, and I'm doing Taylor and Maclaurin series at the moment, and this is the last step of a problem, but i don't ...
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4answers
766 views

Derivation of the Boltzmann factor in statistical mechanics

I have seen similar derivation of the Boltzmann factor many times before, (http://micro.stanford.edu/~caiwei/me334/Chap8_Canonical_Ensemble_v04.pdf , just for example), which I think is incomplete. ...
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1answer
38 views

I like to know if there is approximated expression of something

I like to know if there is approximated expression of below things. $$e^{-x}\sum^{9}_{k=0}\frac{1}{k!}x^{k}$$ $x$ is not always small. I know behind part is Maclaurin series but summation range is ...
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35 views

Regarding a proof in Tu's 'Introduction to manifolds'

While reading Tu's differential geometry book, I came across a theorem which makes the following claim: Let $f$ be a $\mathcal{C}^\infty$ function on an open subset $U\in \mathbb{R}^n$, let $p\in U$, ...
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20 views

Big 'O' Notation - Taylor Series

Q) Use the Taylor Series Expansion to show the first derivative f '(x) can be approximated by $$-(3f(x) -2f(x+h) - f(x-h) / h ) $$ What is the precision? Now I found after using the Taylor ...
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1answer
30 views

Question Regarding Proof of Taylor Remainder Theorem in Tu's “An Introduction to Manifolds”

The statement: Let $f$ be a $C^{\infty}$ function on an open set $U\subseteq \mathbb{R}^n$ which is star shaped with respect to a point $p=(p^1,...,p^n) \in U$. Then there are functions $g_1(x),...,...
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2answers
18 views

Can the original function be derived from its $k^{th}$ order Taylor polynomial?

Coming from a statistics background, I'll provide an example related to fitting a model to an analysis dataset. Let's suppose I suspect the relationship between the mean value of the outcome variable (...
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1answer
19 views

First-order Taylor series expansion

I have a first-order equation that is supposed to be solved using the Frobenius method. I am having some difficulty since the equation is not equal to zero. I would appreciate any help. y' + (1 - x^...
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15 views

Approximating a quadratic term in the constraint set as 2nd order Taylor expansion

I have an optimization problem in the following form: $$\min_{x,y} f(x)+g(y)$$ $$s.t.$$ $$Ax+h(y)=0$$ where $h(y)$ is a quadratic in $y$. Instead of solving this problem directly, $h(y)$ is ...
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2answers
20 views

If $f$ and $h$ are differentiable in $a$ and $h'(a)=f'(a)$. Then $g$ is differentiable and $g'(a)=f'(a).$

Let $f,g,h:X\to\mathbb{R}$ such that $f(x)\leq g(x)\leq h(x)$ for all $x\in X$. If $f$ and $h$ are differentiable in $a$ and $h(a)=f(a)$. Then $g$ is differentiable and $g'(a)=f'(a).$ How can I can ...
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4answers
115 views

What does it mean that a taylor series generated for a function f(x) doesnt converge to f(x)?

If a some function f(x) is continous and has derivatives of all orders on some interval I, and assuming that f(x) can be expressed as a power series on I. And now you generate a taylor series for f(x),...
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1answer
33 views

Order of remainder term in Taylor series approximation

I'm having trouble verifying a bound on the remainder term of a Taylor series approximation. I have a $C^\infty$ function $f$ of compact support. Using the two-term Taylor series for $f$ centered at $...
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46 views

A fast converging limit for $\ln x$ (or why $\ln 2 \approx \sqrt{\sqrt{42}-6})$

I've read a lot about approximating logarithms recently, and apparently it's not easy. It can be done by Taylor series (slow convergence), by continued fractions (also slow) and also by some limits. ...
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2answers
111 views

Series expansion for integral including error function

$\DeclareMathOperator{\erfc}{erfc} \DeclareMathOperator{\Ei}{Ei} $ What is the series expansion of $f$ for small $q$? \begin{align} U(q) &= q e^{q^2}\erfc q\\ I(q,q') &= \int_0^{2\pi}...
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4answers
113 views

Finding out a limit using Taylor series.

So the limit is the following: $$\lim_{x \to 0}{\frac{x^2-\frac{x^6}{2}-x^2 \cos (x^2)}{\sin (x^{10})}}$$ Expansions for $\sin(x)$ and $\cos(x)$ are given: $$\sin x = x-\frac{x^3}{3!} + \frac{x^5}{...
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2answers
42 views

Is $f(x)=\sum_{n\geq 1}\frac{(-x)^n}{n^2+1}$ convex at $x=0$?

Let $\sum_{n=1}^{\infty}\frac{(−1)^n}{ n^2+1} x^n$ be the Taylor series of $f(x)$ about $0$. Then, is it that, $f(x)$ is concave up at $x = 0$?
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1answer
48 views

Estimating $\int_0^{\sqrt 2 / 2} \sin (x^2) dx$ with Taylor Series

I seem to be having trouble with part of this question (Reference: Apostol Volume 1, Section 7.8, Question 8). The full question states: (a) If $0 \leq x\leq \frac{1}{2}$, show that $\sin x = x - x^3/...
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Hermite Expansion of Probability Density Function

While reading this paper by Ait-Sahalia I got stuck with a formula which is quite important. Nevertheless it is not derived explicitly by the author. I resume here the main steps, it is quite a long ...
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19 views

Taylor expansion of the inverse of a function

The Taylor expansion of a function is given by $$ f(x) \approx x^{(1)} + x^{(2)} + x^{(3)} $$ From this we can establish $$ \frac{1}{f(x)} \approx \frac{1}{x^{(1)} + x^{(2)} + x^{(3)}} $$ Is there a ...
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1answer
56 views

Taylor expansion of $\frac1{\|x-y\|}$

Let $0\neq y\in \mathbb{R^3}$ define a function $f$ on $\mathbb{R^3}$ as $$ f(x) = \frac1{\| x-y\|} $$ What are derivatives of $f$ in zero? Or equivalently, what is the Taylor series of $f$ at ...
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1answer
31 views

Can someone draw a plot for this function?

Can someone draw a plot for this function? $ f(x) = \begin{cases} \sum_{i=2}^{x}\left(\frac{\prod_{k=1}^{i-1}\left(2k-1\right)\,\cdot\,-\left(-\frac{1}{2}\right)^{i}}{i!}\right) + \frac{3}{2} & x ...
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2answers
57 views

If $|f(x)| \leq 1$ and $|f''(x)| \leq 1$, show $|f'(x)|\leq 2$

Given $f : \mathbb{R} \to \mathbb{R}$, such that $f'(x)$ and $f''(x)$ exist for all $x \in \mathbb{R}$ and for $x \in [0,2]$, the inequalities $|f''(x)| \leq 1$ and $|f(x)| \leq 1$ hold, I am asked to ...
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Algorithms for Taylor Expansions

Is anything known about fast algorithms for taking symbolic Taylor expansions? I have a homegrown algorithm, but it seems to be exponential in the number of terms requested when operations like the ...
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1answer
20 views

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$.

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$. Now I know that $b_n=\dfrac{f^{(n)}(5)}{n!}$. I have tried various things but I think there is something wrong with my ...
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2answers
16 views

Calculating Taylor series of complex function

I'm going through a past exam paper and found a question I can't do. The question is to write down the Taylor expansion of $\frac{z^2}{z-2}, z \in C$ \ {2}, on the disc $|z| < 2$ I've been ...
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1answer
39 views

Why does $\frac{\frac12 x+\frac18x^2+O(x^3)}{\frac12x-\frac18x^2+O(x^3)}=1+\frac12x+O(x^2)$?

I was reading the solution to a limit through Taylor expansion but did not understand this passage: $$g(x)=\frac{1-\sqrt{1-x}}{\sqrt{1+x}-1}=\frac{\frac12 x+\frac18x^2+O(x^3)}{\frac12x-\frac18x^2+O(...
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1answer
29 views

When does a function admit a Taylor development in 0?

I don't remember much about Taylor series, but in a theorem, I have that something is true if a the function admits a taylor development in x = 0. Is that equivalent to saying that the function in x =...
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1answer
37 views

Proving that a function is real-analytic

I try to solve the following exercise: Let $f:\mathbb{R}\to\mathbb{R}$ with $f(x):=\frac{1}{1+x^4}$. Prove that $f(x)$ is real analytic and compute the radius of convergence of it's Taylor series at ...
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1answer
55 views

Taylor series has zero convergence radius?

Let $$f(x):=\sum_{n=0}^{\infty} \frac{f^{n}(0)x^n}{n!}$$ where the $$|f^{n}(0)| \le C\frac{\Gamma(\frac{n+1}{\alpha})}{\alpha^{\frac{n+1}{\alpha}+1}}$$ for a constant $C>0$ and $\alpha>0$. Does ...
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1answer
29 views

Second Order Accurate Interpolation

On a grid I am having the values of a physical quantity say for example Temperature, at the E,W,N,S and P node all of them being calculated using a second order discretization scheme. I want a second ...
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1answer
31 views

$L^2$ convergence of Taylor series of a holomorphic function

I am reading Otto Forster's book "Lecture on Riemann surfaces" and on pages 109-110, he introduces the space $L^2(D,\mathcal{O})$ of holomorphic square-integrable functions $f:D\to \mathbb{C}$ (where $...
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25 views

Can we derive a Taylor formula for real-valued Fréchet differentiable functions on a normed space?

Using the Lagrange form for the remainder, Taylor's theorem can be stated as follows: Let $I\subseteq\mathbb R$ be an interval, $f\in C^{n+1}(I)$ for some $n\in\mathbb N_0$ and $s,t\in I$ $\...
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0answers
25 views

Taylor expansion of arctan(1+x)

Good evening! I began to learn about Taylor expansions and have to manage such a topic for Monday. I'm looking for the Taylor expansion at the second order: $DL_2(0)$ of $f(x)=\arctan(1+x)$. Having ...
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1answer
45 views

Find the first two terms in the perturbation expansion of the solution

I want to find the $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ terms in the pedestrian expansion $y = y_0 + \epsilon y_1 + \epsilon ^2 y_2 + \dots$, where $y$ satisfies the following second order ODE:...
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2answers
34 views

Prove $\log u > \frac{u - 1}{u}$ for $u > 1$

How to prove that for $u > 1$ $$\log u > \frac{u - 1}{u}$$ without using integrals? I think I'm supposed to use derivatives or Taylor's theorem, as the exercise comes from a lecture about these ...
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1answer
53 views

A Taylor Expansion of a Stochastic Process

As part of a binomial model of a stochastic process, my professor claims that the Taylor Expansion of: $$x\pm = 1 \pm (e^{\sigma^{2}h} - 1)^{1/2}$$ is: $$x = 1 \pm \sigma \sqrt h + O(h^{3/2}) $$ ...
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1answer
35 views

How would you use Maclaurin Series in this question? [closed]

How would you solve $\lim_{x\to0} \frac{1-\cos(x)}{x^2}$ using MacLaurin series?
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1answer
13 views

MacLaurin Series with 2 variables - error

I have a real function $f(x,y)$, where $x,y$ are real. For a fixed $x_{0}$ I want to expand $f(x_{0},y)$ in $y_{0}$ in a first order MacLaurin series How should I write the error with the big O ...
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2answers
70 views

Find a function for the infinite sum $\sum_{n=0}^\infty \frac{n}{n+1}x^n$

I need to find a function $f(x)$ which is equal to the sum $$ \sum_{n=0}^\infty \frac{n}{n+1}x^n, $$ for every $x\in \mathbb{R}$ for which the series converge. Now, using WolframAlpha, I've found the ...
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2answers
98 views

How to calculate the series $-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\frac{1}{10}…$?

$-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\frac{1}{10}...$ After rearrangement the series looks like $\sum^{\infty}_{n=2}\frac{(-1)^{n+1}}{n}$. My way of doing this is using Taylor series of ...
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1answer
33 views

Evaluating limits via Infinite Series

I am to evaluate the following limit of sums and quotients of infinite series $\lim\limits_{z \to 0} \frac{(z^3 + z^6 - z^9 + ...)+(2z^3 -2 z^5 + 2z^7 - 2z^9 ...)}{z^8 + z^{16} + x^{24} + ... }$. I ...
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1answer
12 views

Show that monotonicity implies positive definiteness of the Jacobian

Given $f: \mathbb{R}^n \to \mathbb{R}^n$, $f$ differentiable, $x,y, p \in \mathbb{R}^n$, show that $(x-y)^T(f(x) - f(y)) \geq 0 \Leftrightarrow p^TDf(x)p \geq 0, \forall p \in \mathbb{R}^n$ This ...
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2answers
58 views

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$?

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$? So I think we expand to $x^2$ since the lowest term for $\ln(1+x)$ is $x$ Let $u=\arctan{(x)}$ $\lim\...
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1answer
41 views

Taylor Series for $f(x)$

A function $f$ is defined as $$ f(x) = \left \{\begin{aligned} &{cosx-1\over x^{2}} & for\,x \neq 0\\ &{-1\over 2} & for\,x=0 \end{aligned} \right. $$ Using the first three non zero ...
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2answers
26 views

How to compute $\sum^{\infty}_{n=2}\frac{(4n^2+8n+3)2^n}{n!}$?

How to compute $\sum^{\infty}_{n=2}\frac{(4n^2+8n+3)2^n}{n!}$? I am trying to connect the series to $e^x$ My try: $\sum^{\infty}_{n=2}\frac{(4n^2+8n+3)2^n}{n!}=\sum^{\infty}_{n=2}\frac{(2n+1)(2n+3)2^...
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1answer
10 views

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$?

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$? We know $\sin{x}=\sum^{\infty}_{n=0}\frac{(-1)^nx^{2n+1}}{(2n+1)!}$. Let $t=x-\frac{\pi}{4}$, then $t+\frac{\pi}{4}=x$ Then $\sin{...
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1answer
32 views

How to appraoch solving this series?

I am given the following series and asked to solve it. $\sum\limits_{n=1}^\infty \dfrac{(-2)^n}{(2n+1)!}$ I recognize that this series is somewhat similar to the Taylor series for $sinx$ which is $\...