Tagged Questions

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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0
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1answer
16 views

Convergence of Taylor series about centre of open disc for analytic function.

I define a function on an open set of the complex plane to be analytic if about any point $z_0$ in that set it can be expanded as a power series in $(z - z_0)$ that converges in some neighbourhood of ...
0
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0answers
6 views

Taylor Polynomial to estimate solution of MVC differential equation

I have read a few examples, that you're using derivatives at different points to estimate a polynomial but I need a bit a of guidance to understand how this would work in a multivariate calculus ...
0
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1answer
40 views

Use of taylor series in convergence

Homework problem here, would appreciate an explanation to the answer of this question. Problem: Find the rate of convergence of $$ \lim\limits_{h \to 0} \frac{\sin(h)}{h} = 0 $$ The book solves ...
0
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1answer
54 views

Sum of Taylor Series

I have the converging series: $$ 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+... $$ and I'm trying to find its sum when x = .9. I know this is the Taylor series for some function$f(x)$, and that I can ...
0
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1answer
30 views

Infinite series expansion of $\arcsin (x)$ and $\arccos (x)$

How to find the infinite series expansion of $\arcsin (x)$ and $\arccos (x)$?
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0answers
25 views

Taylor Series Expansion

PROBLEMS ax^2 + x + 1 = 0 (1) 1. Using a Taylor series expansion express the solution to the quadratic equation in Equation (1) as a series. Include terms up to cubic order. Find the cubic term in ...
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0answers
17 views

Is there an expression for $\exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = $?

Does an expression for $$ \exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = ? $$ exist? For j=1 we have the usual expression for translation and scaling $$ \exp\left( t \partial_z\right) f(z) = ...
3
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4answers
48 views

Taylor polynomial of $\frac{1}{2-x}$

Can someone show how to find the Taylor polynomial of $\frac{1}{2-x}$? I tried this: $\frac{1}{2-x}=\frac{1}{1-(x-1)}$ and then use that $ \ T_n(\frac{1}{1-x})=1+x+\dots +x^n.$ But this gives ...
1
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0answers
31 views

Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
1
vote
1answer
33 views

Series representation of function with fractions, logarithms, squares and cosines.

I'm looking for a series representation for $$\dfrac x{x^2+(\log \cos x)^2}$$ Where $x\in(0,\pi/2)$ Note: Both finite and infinite series are accepted. I have tried taylor series, but it requires ...
4
votes
3answers
181 views

How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?

In an older question here in MSE I've asked for the term for the "slicing" of a power series in partial series and have learned that it is "multisection". I' ve been looking at the behaviour of the ...
2
votes
4answers
158 views

How can I accurately compute $\sqrt{x + 2} −\sqrt{x}$ when $x$ is large?

How can the values of the function $f(x) = \sqrt{x + 2} −\sqrt{x}$ be computed accurately when $x$ is large? I have tried using Matllab. I am not able to understand when $x$ will be large.
1
vote
1answer
55 views

Differentiability of the remainder in Taylor's theorem

Suppose we have a function that's differentiable $m$ times over $[a,b]$, we have $a< \alpha < x < b$ and $n < m$. Then $$ f(x) = \sum_{i = 0}^{n-1} \frac{f^{(i)}(\alpha)}{i!}(x - ...
6
votes
4answers
196 views

Closed-forms of infinite series with factorial in the denominator

How to evaluate the closed-forms of series \begin{equation} 1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, ...
1
vote
1answer
16 views

Estimating the remainder for Mac Laurin's series

I'm practicing Taylor's series and i found some old task. Calculate value of function $f(x) = e^x + e^{-x}$ at point $x = \frac{1}{\sqrt 2}$ with error not greater than $d=\frac{1}{20}$ So here's ...
0
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1answer
31 views

Maclaurin series for $\frac{1}{|1+x|}$

I believe that there is no Maclaurin Series for $\frac{1}{|1+x|}$ as the latter is not differentiable at $x=-1$. However, would it be appropriate for me to refer $\frac{1}{|1+x|}$ as 'not a smooth' ...
6
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3answers
138 views

Use Taylor Series method to solve $y''-2xy+y=0$

I am doing some practice problems for solving second order ODEs, and I am a bit stuck on this one. Here is what I have: $y''-2xy'+y=0$ Let $y = \sum_{n=0}^{\infty} C_nx^n \implies y' = ...
1
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0answers
22 views

Taylor's remainder in a compact

My impression is that each function enough regular ($C^\infty$ ) in a compact is equivalent to a polynomial. Is this true? Is there a way to prove it? The expression of the Taylor's remainder just ...
1
vote
2answers
19 views

Gradient decent using Taylor Series

I'm reading a book about Gradient methods right now, where the author is using a Taylor series to explain/derive an equation. $$ \mathbf x_a = \mathbf x - \alpha \mathbf{ \nabla f } (\mathbf x ) $$ ...
2
votes
1answer
69 views

Showing that if the $n$th derivative of a function is bounded then it is real analytic

I reproduce from my lecture notes: Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$. Proof. ...
2
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1answer
46 views

$f$ differentiable and $f(0)=f(1)=0$. , prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$

Let $f$ be differentiable on $[0,1]$ and $f(0)=f(1)=0$. Also, we know $|f''(x)| \le A$ on $(0,1)$, prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$ I'm guessing I should use taylor ...
1
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0answers
30 views

What is the bound on the error that is given by Taylor's inequality?

I have this problem here that I'm not sure how to solve. Consider the polynomial $1 - \large\frac {x^2}{2!}$ as an approximation to $\cos(x)$ on the closed interval $-1 \le x \le 1$. What is the best ...
0
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0answers
86 views

Taylor series expansion and Laplace transform final value theorem

I cant figure out how some transformations are made in one article on physics. Here is expression in s-domain and they want to find its asymptotic value. $$ \xi(s) = \nu_1(s+1)=\frac{1}{(s+1)} ...
2
votes
1answer
35 views

Evaluating $\ln(\cos x))$ using Taylor expansion

Evaluate $\ln(\cos x)$ at $x_0=0$ and with the order of $n=4$. Noticing that $\ln(\cos x) = \ln(1+ \cos x - 1)$ we can use $\ln(1+x)$ Taylor series. Now, I've read I should use: $$\ln(1+x) = x - ...
2
votes
1answer
35 views

Using Taylor series with remainder

Arfken and Weber expand $(1+n^{-2})^{-1}$ as $$(1+n^{-2})^{-1}=1-n^{-2}+n^{-4}-\frac{n^{-6}}{1+n^{-2}}$$ However, if I use Taylor/Maclaurin expansion $$(1+x)^m=1+mx+ \frac{m(m-1)}{2!} x^2 + \cdots + ...
1
vote
1answer
78 views

What is the limit regarding $a$

What is the limit of : $$ \lim_{x\to 0} \frac{\sin(ax) - \ln(1-2x)}{e^{ax}-1-2x-2x^{2}}$$ I did this with Maclaurin, because my exam is about solving these with MacLaurin. Gave $$\lim_{x\to 0} ...
1
vote
1answer
32 views

Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
1
vote
1answer
88 views

Product of two $2$-variables Taylor series

Using the standard multi-index notation, suppose we have the two Taylor series $$ f(\theta) := \sum_{|\alpha|=0}^{\infty} a_{\alpha} \theta^{\alpha} $$ and $$ g(\theta) := \sum_{|\alpha|=0}^{\infty} ...
1
vote
1answer
50 views

What can be said about an infinitely differentiable function whose Taylor series diverges?

What can be said in general of an infinitely smooth function whose Taylor series diverges? According to Borel Theorem it is possible to construct such but what kind of property have those special ...
1
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0answers
46 views

Power series and Taylor series

Let $f:\Bbb R\to\Bbb R$ be a $\cal C^\infty$ function. Consider the power series $$\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$$ and call $R$ its radius of convergence. Then, is it true that in ...
0
votes
1answer
80 views

Limiting the error using Taylor's formula with Lagrange's Remainder

I get stuck on problems about approximating values with Taylor's formula when it comes to determining the amount of taylor terms (n) required to satisfy a minumum error by just using Lagrange's ...
0
votes
2answers
30 views

Find the Taylor series and prove it converges using the defintion

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems. Find the Taylor series about $x=2$ for the function $f(x) = x^5 - ...
2
votes
1answer
40 views

taylor series expansion, derivatives not continuous

As a part of an excercise I am supposed to find the Taylor series expansion for $(1-t)^{\frac{1}{2}}$ on $[0,1]$. According to the remainder theorem: ...
1
vote
1answer
23 views

Maclaurin series - Approximation and interval of convergence

This is a problem which I should apparently be solving with Maclaurin series, but I failed to do so. So I attempted it with binomial series, with 5 terms and an error less than the requirement in ...
1
vote
1answer
26 views

Taylor series of an analytic function that maps the unit disk surjectively onto the upper half plane

Given only that $f(z)$ is analytic and maps the unit disk $|z| < 1$ surjectively to the upper half plane $\Im(z) > 0$, how much can we deduce about $f(z)$? In particular, can we find the radius ...
4
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0answers
58 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
1
vote
1answer
58 views

Lagrange remainder vs. Alternating Series Estimation Theorem: do they always give you the same error bound?

Given a function and its nth degree Taylor series approximation, we can use the Lagrange form of the remainder to get a maximum value of the error of approximation. If the series is also an ...
1
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0answers
90 views

A function equal to its Taylor series on an interval is also equal to its Taylor series on a subinterval with different center

Suppse the power series $ \sum_{n=0}^\infty a_n (x-a )^n$ has positive radius of convergence $R$ and thus defines a real analytic fuction $f$ on $(a-R,a+R).$If $x_0$ is a point with $|x_0-a|<R,$ ...
0
votes
2answers
66 views

Taylor series $(x+1)^{\frac{1}{3}}$

Complete the Maclaurin polynomial of degree three for $(x+1)^{1/3}$. I have completed the first two derivatives of this function and thus have coefficients but am not certain how to put them into ...
0
votes
2answers
48 views

Convergence of general Taylor series

For any $a,h \in \mathbb{R}$, how can we see the series, $\sum_{k=0}^{\infty} \dfrac{f^{(k)}(a)}{k!} h^k$ converges to $f(a+h)$?
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2answers
44 views

When to Taylor expand in a differential equation

I've come across a differential equation: $\dfrac{d\theta}{dx} = f(\theta)$, whose analytic solution is very complicated, but in this situation it is valid to Taylor expand functions around $\theta ...
3
votes
2answers
95 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
1
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2answers
46 views

Estimate eigenvectors of symmetric matrix with almost vanishing diagonal

Is there a way to approximate the eigenvectors of a symmetric matrix with almost vanishing diagonal elements, i.e. with the block matrix form, \begin{equation} M=\left( \begin{array}{cc} ...
0
votes
0answers
13 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...
0
votes
1answer
27 views

Taylor's Theorem in 2 variables

I've never been 100% happy with using taylors theorem, mainly because I see it used in a bunch of different ways and I'm never sure in which situations it is valid. The way I was introduced to it was ...
2
votes
2answers
37 views

Evaluating a limit with Taylor Series

I would like to find the following limit using Taylor Series: $$\lim_{x\to0}\frac{6\sinh x-6x-x^3}{x^4(6+x^2)\sinh x}.$$ Now my question is the following: How do I know exactly how to approximate ...
0
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0answers
31 views

Lagrange's form of the remainder vs Cauchy's form

So far (while practicing exercises) I've used Lagrange's form of the remainder. Is there a situation when Cauchy's form comes in handy while Lagrange's form fails for some reason? Is there a rule of ...
2
votes
0answers
72 views

How to express $e^{yS^2}(f(x))$ in closed form where $\frac{d}{dx}=S$

$$ f(x)+\frac{y.f'(x)}{1!}+\frac{y^2 f^{''}(x)}{2!}+\cdots=e^{yS}(f(x))=f(x+y) \text{ where }\frac{d}{dx}=S$$ is a operator $$ f(x)+\frac{y.f''(x)}{1!}+\frac{y^2 ...
4
votes
1answer
148 views

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
1
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0answers
52 views

Given a function $f$, find the largest $n$ such that $f(x)/x^n$ can be defined at $x=0$ to become differentiable there

Let $f(x) = \ln\left(\frac{x^2}{2}+1\right)+\cos x -1$. Find the largest $n\in\Bbb{N}$ such that there is $C\in\Bbb{R}$ such that: $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C ...