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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65 views

Where to stop a taylor expansion of a function of more than one variable?

I tried to taylor expand a function of three variables $f(x,y,z)$ around $y=0,z=0$ but I don't find a way to decide where to stop the expansion. Usually when one expands a function of one variable ...
3
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2answers
101 views

Show that $f(x)=e^x$

In this case $f(x)=1+x+x^2/2!+x^3/3!+x^4/4! + ... = \sum_{n=0}^\infty \frac{x^n}{n!}$. I understand it conceptually in terms of the Taylor series, but I have no idea how to prove it rigorously.
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1answer
26 views

Error estimation help

I'm supposed to find a Taylor polynomal of the $n^{\text{th}}$ degree, where $x = a$, and estimate the error for the given interval. The problem I'm given is: $$f(x) = \sqrt{x}, a = 4, n = 2, 4 \leq x ...
2
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1answer
65 views

Show that $|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$ and $M_1 \le 2\sqrt{M_0M_2}$

Let $f$ be a twice derivable function and $M_i =\sup_{x \in \mathbb{R}} |f^{(i)}(x)|$ and $|M_0|, |M_2|<\infty $. Preferably using the Taylor series on the interval $[x,x+h]$ show the following ...
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0answers
17 views

Why more smooth the function the more precise finite difference method?

As the title, Why more smooth the function the better finite difference method? I guess that if the function is smooth we can better approximate with Taylor series, but formally how this helps? ...
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1answer
108 views

Finding an upper bound error of a Maclaurin polynomial.

Using a 3rd order Maclaurin polynomial, find an upper bound on the error when log(1+x) is approximated by a 3rd order polynomial for |x|<= 0.1. For some reason I keep getting a different answer ...
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1answer
56 views

Prove the taylor series of $ \cos(2z)$

First i turned $$\cos(2z) = \frac{e^{2iz} + e^{-2iz}}{2}$$, then using the taylor series of $$e^{z}$$I calculated the taylor series of both arguments. $$\frac{e^{2iz}}{2} = \sum_{n=0}^{\infty ...
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1answer
100 views

Question about infinitely many times differentiable function.

Could you please give me some hint how to solve this problem: Suppose $f(x)$ is infinitely many times differentiable function on R, $f(0)=f'(0)=f''(0)=0$. Prove : for all $A>0$ exists some ...
2
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0answers
25 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
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4answers
84 views

Taylor Series - general 'what' and 'why' questions

I am a little confused with the Taylor Series at the moment, so please forgive me for my very basic questions. If we were to approximate a function, say $cos(x)$, I let $f(x)=cos(x)$ And I have been ...
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1answer
63 views

Why is the series expansion of $\sin(x)$ at $x=n\pi$ different to the expansion of $\sin(2x)$ at $x=\frac{n\pi}{2}?$

I'm trying to do a series expansion of $\sin(2x)$ about the point $x=\frac{n\pi}2$ where is an integer. I thought that the expansion would be the same as for $\sin(x)$ about the point $x=n\pi$ but ...
0
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0answers
89 views

Taylor theorem remainder term

I'm having trouble applying the formula for the remainder in the Taylor's theorem. From Wikipedia we know that for $f(x)=f(a)+f'(a)(x-a)+…\frac{f^{(n)}(a)}{n!}(x-a)^{n}+R$ the remainder $R$ in the ...
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1answer
128 views

L'Hospital's rule vs Taylor series

One classical application of Taylor expansions is to obtain polynomial equivalents of complicated functions and use them to compute limits. For example, with Landau notations, we have ...
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0answers
39 views

Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
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3answers
41 views

Find the first three terms of the taylor expansion of $\frac{cos(z)}{1 + z^2}$

The question is: Find the first three non zero terms for the taylor series for $\frac{\cos(z)}{1 + z^2} $ around $z_0 = 0$ What I've done so far is let $f(z) = \frac{\cos(z)}{1 + z^2}$ Then I let ...
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2answers
38 views

Complex functions and Taylor series

Find the Taylor series arround $z_0=0$ write radius of convergence a) $f(z)=\cosh(z)$ b) $f(z)=\log(z+1)$ I don't know how it works with the complex functions. Could you show me the workflow? I ...
4
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2answers
82 views

Is $\cos(\frac{\pi}{3})$ exactly equal to 0.5 or approximately equal to 0.5

We know that $\cos(\frac{\pi}{3})=\frac{1}{2}$, but the expansion for $\cos(x)$ is as follows: $$ \cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$$ So this would make $$\begin{align} ...
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3answers
63 views

Proving an inequality using Taylor's Theorem

I need to show that $ x^{1/3} < \frac{1}{3}x + \frac{2}{3} \forall x \in (0,1)$. I have been given the hint to consider the expression $\frac{1}{3}x - x^{1/3}$, but the Taylor Series centred at ...
6
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2answers
269 views

Can we prove that all zeros of entire function cos(x) are real from the Taylor series expansion of cos(x)?

Q1: Can we prove that all zeros of cos(x) are real from the following Taylor series expansion of cos(x)? $$ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^k}{(2k)!}x^{2k} $$ The Riemann $\xi(z)$ function is ...
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1answer
68 views

How do I compute the Taylor Series for $\arctan(x)$?

I've just stumbled upon Taylor Series on Wikipedia and I've been trying to obtain an expansion for $\arctan(x)$, but I can't manage to see a pattern for the $n$th derivative . Can someone come up with ...
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3answers
97 views

Taylor development of $\arctan(\cos(x))$ near $0$

How would I find the "Taylor development of $\arctan(cos(x))$ near $0$ at order $5$?" I am translating that from french, so I am not sure how I have to call it it english. By order $5$ I mean that I ...
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0answers
64 views

Wynn-epsilon convergence

How could I use the Wynn-epsilon alghoritm in Matlab to accelerate the convergence of a Maclaurin series? I want to extimate the first derivative of $f(x)$, so $$f'(x)= \sum_{k=0}^\infty ...
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1answer
2k views

Find the Maclaurin series for $\cos(2x)$ using the series for $\sin(2x) $.

I know that $$\sin(2x)= 2x - \frac{8x^3}{3!} + \frac{32x^5}{5!} - \frac{128x^7}{7!} + \cdots $$ $$\sin(2x)= \sum_{n=0}^\infty (-1)^n {2^{2n+1}x^{2n+1} \over (2n+1)!}$$ But I don't see how I can use ...
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0answers
30 views

What is the radius of convergence of the derivative of a smooth Taylor series?

On this website I found that the derivative of a Taylor series has the same radius of convergence as the Taylor series itselves. However, there is no reference added, and I seem to be unable to find ...
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0answers
30 views

Finding a function with a given Taylor expansion

Is there any function $f(x)$ which has the following Taylor series representation? $$ f(x) = \sum_{k=0}^{\infty}{c_{k} (1 + \frac{2x^{2}}{k})^{-k/2}}. $$ for some coefficients ...
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2answers
32 views

Taylor theorem and a $C^{3}$ function with the following property…

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^3$ with $$f(a+h)=f(a)+f'\left(a+\dfrac{1}{2}h\right)h$$ whenever $a \in \mathbb{R}$ and $h \geq 0$. By applying Taylors Theorem to $f$ and to ...
4
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2answers
557 views

Taylor polynomial with Lagrange remainder

In my course there's a paragraph: Taylor polynomial with Lagrange remainder, The paragraph starts with a theorem (I left out the constraints): $$ ( \exists \theta \in ]0,1[)(f(a +h) = T_{f,a,n}(a + ...
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1answer
1k views

Computing the elements of a Hessian matrix with finite difference

I have a generic function $g(x)$ where $x$ is an 6-dimensional vector. Now I want to compute the Hessian of this function for a point $x_0$. What is the most efficient way to do this? Can I do this ...
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2answers
57 views

Taylor Polynomial converges to the original function?

If $$P_n(x)=x-\frac{x^2}{2}+\frac{x^3}{3}-..+\frac{x^{2n+1}}{2n+1}$$ (It's taylor series of $\ln(1+x)$ near x=0. Then can I say that: $\lim_{n\to\infty}{P_n(x)}=\ln(1+x)$, please explain why or why ...
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0answers
34 views

Plank's first law expansion

I'm getting a little stuck on this question. The question is: show that for $KT \gg h\omega$, the first law of Planck: $\displaystyle U =\frac{h\omega}{e^{(h\omega/KT)}-1} \approx KT - ...
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2answers
49 views

Taylor Polynomials — $\cos(x)$ [closed]

Show that $$\forall x : \cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots+\frac{(-1)^nx^{2n}}{(2n)!}$$ I know that this is true because it is one of the most common Taylor polynomials. ...
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2answers
75 views

Taylor series of a division-by-zero equation

I need to calculate taylor series of $(\frac1{t^3}+\frac3{t^2})^{1/3} - \sqrt{(\frac1{t^2}-\frac2{t})}$ at $t = 0$ to calculate limit $(\frac1{t^3}+\frac3{t^2})^{1/3} - ...
5
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2answers
133 views

Remainder form of Taylor polynomial at $x_0$: $ \frac{1}{n!}f^{(n)}(x_0 +\theta(x-x_0))(x-x_0)^n$ with $\theta \to \frac{1}{n+1}$ as $ x \to x_0$

If the function $f: \mathbb R \to \mathbb R$ is $n+1$ times differentiable at $x_0$ and $f^{(n+1)}(x_0) \neq 0$, then a form of the remainder in Taylor's Formula is supposedly $$r_n(x_0;x) = ...
4
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1answer
3k views

Prove Taylor expansion with mean value theorem

On http://vapour-trail.blogspot.com/2006/03/brief-explanation-of-taylor-series-via.html one can find an hint at how to derive Taylor expansions from the mean value theorem. The process goes as ...
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2answers
126 views

Find the Taylor series generated by f at x=a.

$f(x) = \frac 1 {9 - x}, a = 3$. The answer in the book is $$\sum_{n = 0}^{\infty} \frac{(x - 3)^n}{6^{n + 1}}$$but I'm not sure how to get the above.
1
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1answer
62 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
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1answer
77 views

How is the Harris Corner detector derived from a Taylor Expansion?

It looks more like an assumption about local linearity of the function when the shifts u and v are small. This description of ...
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3answers
92 views

Convergence of Taylor Series

My professor made this claim about Taylor Series convergence in my Complex Variables class and I am still not entirely convinced (he said it's explained in the textbook and textbook states, "we will ...
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0answers
32 views

Somehow “mirroring” the Taylor-expansion of some $g(x)$

In proceeding with an older discussion of a summation-procedure for divergent series using the matrix of Eulerian numbers I came to a rather general formulation but cannot/do-not-know-how-to make ...
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2answers
246 views

Derivation 9.97 in Jaynes' Probability Theory

In page 298 of Jaynes' Probability Theory: the Logic of Science, equation (9.97), Jaynes says: We expect that, if hypothesis $H$ is true, then $n_k$ will be close to $np_k$, in the sense that the ...
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1answer
100 views

Why does $\arctan(\frac{\tan \theta}{2}) \approx \frac{1}{2 - \theta} - \frac{1}{2 + \theta}$ for small $\theta$?

In answering this question, I needed to show that $\arctan \left( \frac{\tan \theta}{2} \right) \approx \frac{\theta}{2}$ when $\theta$ was small. So, naturally, I computed the first few terms of the ...
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1answer
35 views

Taylor series remainder question

Let $f(x)=\frac{\sin(x)}{x}$ when $x\neq 0$ and $f(x)=1$ when $x=0$. Starting with the Taylor polynomial of degree $2n+1$ for $\sin(x)$ and the estimate for the remainder term, show that ...
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0answers
55 views

Analytic Function Derived From Recursive Reverse Taylor Series?

Given the following recursive relation: $a_0 = 1,$ $a_n = a_{n-1}(p-2q)+2(-p)^n$ is there a simple function that has this as its Taylor series, i.e. $f(x) = \sum_{n=0}^{\infty} \frac{a_n}{n!}x^n$ ...
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1answer
33 views

Maclaurin series and taylor

Im trying to find the first four terms of Maclaurin series of $\space0.15t^2$ and evaluate $$\int_0^1 e^{-0.15}t^2 dt $$ Please this is a revision question. How do i go about it?
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2answers
114 views

Taylor Series Substitution $e^{x^2-1}$

If I'm using substitution to find a Taylor series about $x=1$ for $e^{x^2-1}$, but I'm given the Maclaurin series for $e^x$, how come the fact that the Taylor series is about $x=1$ doesn't matter when ...
2
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3answers
276 views

Evaluate the limit with Taylor series

How one can evaluate following limit: $\lim_{x\to\infty} x(\frac{1}{e}-(\frac{x}{x+1})^x)$ ? I've found this exercise in the chapter about Taylor series, but I have no idea how to solve it.
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0answers
37 views

Taylor Remainder proof for $e^x$

Prove that if $x\leq 0$ then the remainder term $R_{n,0}$ for $e^x$ satisfies $|R_{n,0}|\leq \frac{|x|^{n+1}}{(n+1)!}$. First, $P_{n,0}(x)=1+x+\frac{x^2}{2!}+\cdots+\frac{x^n}{n!}$ with ...
1
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1answer
43 views

Taylor series of $e^{(x-1)^2}$ about $x=1$

How would we find the Taylor series of $e^{(x-1)^2}$ about $a=1$? I tried finding the answer using the Taylor series of $e^x$ about $a=1$ which I was able to do correctly. When I substituted ...
0
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2answers
31 views

Maclaurin series for (cosx-1)/(x^2)

The solution for this is -1/x+x^2/4!-x^2/6!......, but I'm not sure how to derive this Maclaurin series from cos x. The solution just divided each term in the Maclaurin series for cos x by x^2, and ...
1
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1answer
112 views

Proving Remainder of Taylor Series of 1/(1-x) approaches 0

It is well known that the Taylor (Maclaurin) series of $f(x) = \frac{1}{1-x}$ is $\sum_{n=0}^\infty x^n$ on $(-1,1)$. I am having difficulty proving the equality of these two. The error term is ...