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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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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0answers
25 views

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 ...
0
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0answers
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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0answers
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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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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1answer
40 views

Does multiplying Taylor series by an integer change the interval of validity.

If I have a Taylor series for example, $\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \ldots, \qquad \text{valid for $-1<x<1$} $ and I multiply the series by some integer, let's say $5$, in ...
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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
15 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 ...
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1answer
36 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
28 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
54 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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0answers
23 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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0answers
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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1answer
30 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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1answer
54 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
28 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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2answers
32 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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21 views

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
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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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
11 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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1answer
32 views

If the derivative is written as shifts, can you relate it to the laplace/fourier tranform?

I was wondering if there is a way to write the derivative as an exponential? This might sound crazy at first, but I recently came across this formula for the Taylor expansion in three dimensions: ...
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4answers
98 views

How to evaluate this limit using Taylor expansions?

I am trying to evaluate this limit: $\lim_{x \to 0} \dfrac{(\sin x -x)(\cos x -1)}{x(e^x-1)^4}$ I know that I need to use Taylor expansions for $\sin x -x$, $\cos x -1$ and $e^x-1$. I also realise ...
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1answer
85 views

Approximate roots of nonlinear equation (non-integer polynomial)

In case of pulsating bubble arising from underwater explosion, bubble radius satisfies the following equation. $x^3\dot{x}^{2} + x^3 + \frac{k}{x^{3(\gamma-1)}} = 1$ The minimum and maximum bubble ...
0
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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
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 $u=\arctan{(x)}$ ...
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2answers
22 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: ...
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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 ...
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0answers
42 views

Is there a faster convergence series than the Taylor series?

I am looking for a series expansion which will converge faster than the Taylor series. I mean $$ f(x)=\sum_{n=0}^{N}\frac{f^{(n)}(0)}{n!}x^n $$ For some function you may need large $N$ to get a ...
0
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2answers
80 views

Verify f'(x) = e^x

The following is a proof I wrote to prove that given $f(x)=e^x$, $f'(x)=e^x$. For this proof we must use the Taylor Series for $e^x$, $\sum\limits_{n=0}^{\infty}\dfrac{x^n}{n!}$. Since the derivative ...
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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 ...
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2answers
41 views

Almost Taylor's Theorem Proof through Integration by Parts

I ALMOST derived Taylor's theorem, which here is $f(x)=\sum_{n=0}^\infty\frac{(x-a)^nf^{(n)}(a)}{n!}$, where $a$ is some arbitrary constant. My attempt: $$f(x)+C=\int f'(x)dx$$ $$\int ...
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0answers
16 views

Is the following correct way of manipulating taylors series?

For $\sum^{\infty}_{n=1}\frac{(-1)^{n}\pi^{2n}}{4^n(2n+1)!}$. Let $x=\frac{\pi}{2}$, the series becomes ...
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2answers
38 views

How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$?

How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$? It should be associated with the geometric series. Setting $t=x-3,\ x=t+3$, then I don't know how to continue, could someone clarify the ...
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1answer
34 views

How to compute $\sin{(\pi x)}$ about $\frac12$ in taylor series?

The correct answer is supposed to be $\sum\frac{(-1)^n}{(2n)!}\pi^{2n}(x-\frac12)^n$ which I don't understand. Since the function is about $x=\frac12$, so $(x-\frac12)^n$ is good. But ...
2
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2answers
108 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') &= ...
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2answers
94 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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3answers
40 views

How to compute $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos{(2x)}-3x^2}{x^2\sin{x^2}}$ limits by using taylor series?

How to compute $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos{(2x)}-3x^2}{x^2\sin{x^2}}$ limits by using taylor series? I think that we need to take every familiar taylor series (i.e. $e^x,\sin{x}$) and ...
8
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2answers
109 views

How can I get f(x) from its Maclaurin series?

I know how to get a Maclaurin series when $f(x)$ is given. I have to find $\sum_{n=0}^{\infty}\frac{f^{(k)}(0)}{k!}x^k$. But how can I get $f(x)$ from its Taylor series? The problem is $$f(x) = ...
2
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1answer
25 views

How to compute the following series using taylor expansion manipulation?

How to compute $\sum^{\infty}_{n=0} \frac{x^n}{(n+2)n!}$ and $\sum^{\infty}_{n=0}(-1)^n \frac{(n+1)x^{2n+1}}{(2n+1)!}$ using taylor expansion manipulation? $1.\sum^{\infty}_{n=0} ...
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0answers
15 views

Maclaurin polynomial expansion of $y$ about 1?

Consider the differential equation $\frac{dy}{dx}=2x+\frac{y}{x}$, where $\frac{dy}{dx}=1$ when $x=1$. Find the first three non-zero terms in the Maclaurin polynomial expansion for $y$ about ...
2
votes
1answer
18 views

How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$ and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$?

How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$ and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$ ? For the first one, using substitution, let $t=x-3$, then $x=t+3$. Then ...
4
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7answers
549 views

How do you create an alternating series with the sign being the same twice in a row?

I am working on a Taylor series question and I have created a series which alternates however, it does so in doubles. in other words it follows the following pattern: $x$, $x$, $-x$, $-x$, $x$, ...
1
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2answers
37 views

Uncertain how the following step was accomplished.

I'm working through a book example that aims to find the first two nonzero terms of the Laurent expansion of $f(z)=\tan(z)$, about $z=\frac{\pi}{2}$. The substitution $z=\frac{\pi}{2}+u$ is made ...
0
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1answer
26 views

Taylor Polynomial Approximations

I am asked to find a Taylor Polynomial approximation accurate to within $10^{-3}$ for the following function $$f(x)=\frac{1}{x+1}, x \in [-\frac{1}{2},\frac{1}{2}]$$ We know the Taylor expansion for ...
0
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3answers
48 views

Use of Taylor series expansion to find second derivative for sixth order method

Use Taylor's expansion to derive sixth order method (i.e $\mathcal{O}(h^6)$) for approximating the second derivative ($f '' (x_0)$ ) for given sufficiently smooth function $f(x)$. I have this things ...