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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39 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
41 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 ...
3
votes
1answer
2k views

First four Taylor Series Expansions

I'm supposed to write the first four taylor series expansions of $f(x=0)$ using: one term, two terms, three terms, four terms This is the function: $$f(x) = x^3 - 2x^2 + 2x - 3$$ Should I be using ...
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0answers
28 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 ...
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0answers
19 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
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1answer
30 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
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1answer
28 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
75 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} ...
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1answer
30 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 ...
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1answer
14k views

Taylor series expansion of arctan(x) around the point 0

This is actually a technical question about why I'm getting different results when trying to do the same thing using Maxima and WolframAlpha. When I enter expand arctan(x) in WolframAlpha I ...
4
votes
1answer
142 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 ...
2
votes
1answer
70 views

Evaluate a limit using Taylor series

Let $$\lim\limits_{x\to 0}\frac{({\ln(1+x) -x +\frac{x^2}{2})^4}}{(\cos(x)-1+\frac{x^2}{2})^3}$$ Now, I know that I should utilize Taylor polynomial. $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...
1
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1answer
37 views

Using Maclaurin approximation to find the limit of $(\ln(1+x^2)-\ln(1-x^2))/(e^{x^2}-e^{-x^2})$ as $x\to 0$

I have this assignment: $$\lim_{x\to0}\frac{\ln(1+x^2)-\ln(1-x^2)}{e^{x^2}-e^{-x^2}}$$ And by using the Maclaurin approximations I get this: ...
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1answer
37 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 ...
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0answers
27 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} ...
3
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2answers
178 views

Next term in $(1+a/n)^n \rightarrow \exp (a)$

Working on the generalized birthday problem, where you draw with replacement from $\{1,2,3, \ldots,d\}$ and look for the number of draws $n$ for which you have greater than $1/2$ chance of a match I ...
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0answers
39 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 ...
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2answers
26 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 - ...
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1answer
78 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 ...
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0answers
22 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: ...
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1answer
132 views

Multiple differentiability from Taylor expansion

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a real function, and let $0\leq n\leq+\infty$. We make the following assumption: For every $a \in\mathbb{R}$ and for $k=n$ (resp., in the case $n=+\infty$: ...
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1answer
21 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 ...
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1answer
18 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 ...
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1answer
27 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 ...
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0answers
41 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 ...
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0answers
83 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 $|x_0-a|<R,$ ...
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2answers
64 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 ...
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2answers
39 views

Two-dimensional Taylor linearisation

I have to perform a first order taylor expansion of a function $f(\vec{x}) = f(x+u,y+1)$ at the point $\vec{a} =(x,y)$. My solution reads $$ f(\vec{x}) \approx f(x,y) + \left( \begin{matrix} ...
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2answers
42 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
41 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 ...
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2answers
93 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
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2answers
42 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} ...
2
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0answers
62 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 ...
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0answers
11 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 ...
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1answer
26 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
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2answers
32 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 ...
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0answers
13 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 ...
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0answers
50 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 ...
2
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1answer
21 views

Manipulating Taylor series with $\operatorname{Log} z$

I'm trying to prove the following: $$ \operatorname{Log}z = \sum_{n=1}^{\infty} \frac{\left(1-\frac1z\right)^n}{n} $$ for $\left|1-\frac1z\right|<1$. Does anyone have advice on where to ...
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1answer
44 views

Suppose $f \in C^{\infty}(\mathbb R)$ and $\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0.$ Show $f$ is analytic on $\mathbb{R}$.

Suppose that $f \in C^{\infty} (-\infty , \infty)$ and that $$\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0$$ for all $a\in \mathbb{R}$. Prove that $f$ is analytic on ...
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1answer
24 views

Laurent series of $f(z)=(z^2-1)\mathrm{cos}\frac{1}{z+i}$ in $z_{0}=-i$ and $Res[f(z), -i]$

This is how I've done so far: $$ f(z)=(z^2-1)\mathrm{cos}\frac{1}{z+i} \\w=z+i \;\;\;\; \Rightarrow \;\;\;\; z=w-i \\f(w)=((w-1)^2-1)\mathrm{cos}\frac{1}{w}=(w^2-2wi-2)\mathrm{cos}\frac{1}{w} ...
3
votes
3answers
60 views

Show $\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$

I'm reading a proof which aim to show that: $$\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$$ the Taylor expansion of $\ln(1+x)$ is (not by chance): $$x - ...
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0answers
32 views

Taylor expansions in two variables

I need help in this proof Can I use Taylor expansion in Algebra? someone could give more detail of this Taylor expansion? Thanks in advance EDIT The main question is how the author get this "Y + ...
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vote
1answer
37 views

Solution of $d^2u/dx^2 + u/A = 0 \ (\text{or } \ C),$ with conditions

Does the following ODE: $$d^2u/dx^2 + u/A = 0 \quad (\text{or } \ C),$$ have a solution with the conditions: $$ \left.\frac{d^2u}{dx^2}\right|_{x=0} = 0, $$ $$u(x=0) = B$$ and $$ ...
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2answers
38 views

Multplication of series

My textbook is taking about the Cauchy product and I don't quite understand it and it says that when multiplying series, the sum of the third one is equal to the product of the sums of first two ...
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0answers
23 views

Relationship between Lagrange interpolation and Taylor expansion

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h_{-1}$ and $x_{1} = x_0 + h_1$ with $h_1, h_{-1}$ > 0. Given a smooth function f, and an approximation to $f'(x_0)$ given by the ...
0
votes
1answer
28 views

Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
1
vote
1answer
49 views

Taylor series and Lagrange's remainder f(x)=$e^x$

In my textbook the Lagrange's remainder which is associated with the Taylor's formula is defined as: $R_{n}(x)= \frac{(x-a)^n}{n!} f^{(n)}(a + \vartheta (x-a))$, for some $\vartheta$ $\in$ <0 ,1> ...
0
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0answers
73 views

Relationship between $\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$ and $\sum\limits_{n=0}^\infty \frac{a_n^2 x^n}{n!}$

For an analytic function with the property $f^{(n)}(0)=a_n$, we have $f(x)=\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$. This can be extended to $f^{(n)}(x)=\sum\limits_{n=0}^\infty \frac{a_{n+1} ...