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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73 views

Taylor Polynomial Proof

I am going over a previous year's test and I have no idea how to approach this question. If anyone could please help. Let $g(x)=e^{x^2}$.
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2answers
109 views

Finding the Maclaurin series

Find the Maclaurin series for $f(x)=(x^2+4)e^{2x}$ and use it to calculate the 1000th derivative of $f(x)$ at $x=0$. Is it possible to just find the Maclaurin series for $e^{2x}$ and then multiply it ...
2
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0answers
80 views

Series expansion of a series

I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. ...
3
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3answers
542 views

Finding the Taylor series expansion of $f(z)=\frac{e^{z}-1}{z}$ around $0$

Find the Taylor series expansion of $f(z)=\displaystyle\frac{e^{z}-1}{z}$ around $0$. I have no idea where to start.
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1answer
468 views

A proof of the the second derivative test?

Suppose $f\in C^3$ in some ball centered at a, where $a\in \Bbb{R}^2$,and $\nabla f=0$ at a, but not all second derivatives of $f$ are zero at a. Show how can local maximums local minimums or neither ...
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2answers
75 views

Question about taylor expansion

If I was given a function which its derivative is bounded for every $x>0$ (means: $|f'(x)|\le M$), How can I prove that $\lim_{x\to\infty}\frac{f(x)}{x^2}=0$?
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1answer
80 views

Taylor polynomials expansion with substitution

I am working on some practice exercises on Taylor Polynomial and came across this problem: Find the third order Taylor polynomial of $f(x,y)=x + \cos(\pi y) + x\log(y)$ based at $a=(3,1).$ In the ...
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1answer
36 views

Polynomial of degree four of $f(x)=\sqrt{x}$

Given $f(x)=\sqrt{x}$ Find a polynomial $P(x)$ of degree three such that $P^{(k)}(4)=f^{(k)}(4)$ for $k=0,1,2,3,4$. I know this has to do with Extended mean value theorem, or, Taylor Formula. ...
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0answers
254 views

Consistency order of backward Euler method

How can I proof that backward Euler method has consistency order 1? Implicit function theorem states that for a sufficiently small $h$, $$ \vec{y}_1 = \vec{y}_0 + h f(t_1,\vec{y}_1) $$ has a unique ...
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2answers
294 views

Stirling's Formula - Comparison Test Method

The following question concerns the convergence of Stirling's Approximation for $n!$ I have $r_n = \frac{\sqrt{n}}{n!}(\frac{n}{e})^n$. I have expressed ...
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2answers
148 views

Taylor expansion with change of variables question.

Find the Taylor polynomial of order 3 of $$f(x,y) = (x - 1)^{2} + \sin(\pi y) + x \ln(y)$$ based at $(x,y) = (2,1)$. So I'm really lazy and don't want to take the derivative of that, so let ...
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2answers
64 views

Prove that $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$ where $|R_8(x)|\leq \frac{x^8}{8!}$

Prove that $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$$ where $|R_8(x)|\leq \frac{x^8}{8!}$
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1answer
84 views

Let $f(x)=e^{-1/x}$ for $x>0$ and $f(x)=0$ for $x\leq 0$. Prove that $f^{(n)}(0)=0$ for all $n\in \Bbb N$.

Let $f(x)=e^{-1/x}$ for $x>0$ and $f(x)=0$ for $x\leq 0$. Prove that $f^{(n)}(0)=0$ for all $n\in \Bbb N$. I'm reading the solution, and I understand how to prove that all derivates must be of the ...
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2answers
63 views

Is it true that if the taylor series of $f$ converges it converges to $f$?

Is it true that if the taylor series of $f$ converges it converges to $f$? So if I want to prove that $\lim R_n(x)=0$, could I prove this using the fact that a given taylor series converges ?
3
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1answer
562 views

Find the taylor series for $\cos x$ and indicate why it converges to $\cos x$ for all $x\in \Bbb R$.

Find the taylor series for $\cos x$ and indicate why it converges to $\cos x$ for all $x\in \Bbb R$. I've posted my own proof, I hope it is correct :)
3
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2answers
809 views

Proof of the “Radius of Convergence Theorem”

I can't figure out how it is valid to invoke the Absolute Convergence Theorem, whose hypothesis is "Let the power series have radius of convergence R", to establish case c of the Radius of Convergence ...
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2answers
798 views

Find the Taylor series for $\sinh(x)$ and indicate why it converges to $\sinh(x)$.

Find the Taylor series for $\sinh(x)=\frac 1 2(e^x-e^{-x})$ and indicate why it converges to $\sinh(x)$.
4
votes
1answer
98 views

Write down the equation of the tangent plane and compute the Taylor series of the function

Set $f(x,y,z) = x + y + z + x^2 + y^2 + z^2$. Consider the surface $$S = \{f(x,y,z) = 0\} \subset \mathbb{R}^3$$ near the origin $o = (0,0,0) \in S$. Write down the equation of the ...
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1answer
90 views

$\frac{\mathrm d^n}{\mathrm d x^n} e^{-\frac {1}{x^2}} = 0$ at $x=0$ [duplicate]

This is an exercise from David Brannan's Mathematical Analysis. I've proved parts (a) - (c) but need help with Part (d). Any guidance appreciated. EDIT I have solved it, by induction using the ...
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3answers
282 views

Let $g(x)=e^{-1/x^2}$ for $x\not=0$ and $g(0)=0$. Show that $g^{(n)}(0)=0$ for all $n\in\Bbb N$.

Let $g(x)=e^{-1/x^2}$ for $x\not=0$ and $g(0)=0$. Show that $g^{(n)}(0)=0$ for all $n\in\Bbb N$. In the text it is already proven that for the function $f$ with $f(x)=e^{-\frac{1}{x}}$ for $x>0$ ...
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1answer
63 views

Taylor series representation of a function.

I'm working on expressing the function $f(x)=\frac{6}{x}$ as a taylor series about $-4$. I've got the general idea, but I'm not quite there yet. I've come up with the equation ...
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2answers
53 views

Find the Taylor series of $\frac{1}{x+1} $ at $x=2$

This is what I did: $\begin{align*} f(x)&=&(x+1)^{-1}\\ f'(x)&=&-(x+1)^{-2}\\ f''(x)&=&2(x+1)^{-3}\\ f'''(x)&=&-6(x+1)^{-4}\\ f''''(x)&=&24(x+1)^{-5}\\ ...
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2answers
2k views

Vector taylor series

Classical Electrodynamics by Jackson says "With a Taylor series expansion of the well-behaved $\rho (\mathbf{x'})$ around $\mathbf{x'} = \mathbf{x}$ one finds ..." and then he says basically that ...
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2answers
382 views

A property of roots of the truncated series for $\sin(x)$

Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$ In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$. ...
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1answer
83 views

How to compute tangent of a discretized curve

I have a discretized curve defined by a 2D matrix $M$ where $M(i,j)=1$ means the point $(i,j)$ is on the curve. For each of these points, I want to calculate its tangent vector by fitting a polynomial ...
2
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2answers
162 views

Maclaurin series for $\frac{x}{e^x-1}$

Maclaurin series for $$\frac{x}{e^x-1}$$ The answer is $$1-\frac x2 + \frac {x^2}{12} - \frac {x^4}{720} + \cdots$$ How can i get that answer?
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4answers
125 views

Taylor expansion of $x^x-1$ around 1

How do I find Taylor expansion of(around 1): $$f(x)=x^x-1$$ The answer should be: $$(x-1)+(x-1)^2+\frac 12(x-1)^3+\cdots$$ How the answer was obtained?
4
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1answer
583 views

Find Maclaurin series of $(\sin(x^3))^{1/3}$

How do I find Maclaurin series for the function: $$\sqrt[3]{\sin(x^3)}$$ The answer should be: $$ x - \frac {x^7}{18} - \frac {{x}^{13}}{3240} + o(x^{13})$$ I tried: $$\sin x = x - \frac ...
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1answer
27 views

Some question about Taylor polynomia

$$f(t) = \frac{100e^t}{10+e^t} \text{for all } t \in \mathbb{R}$$ (1) show that $$\frac{d}{dt}f(t)=f(t) - \frac{1}{100} (f(t)^2)$$ (2) waht is the second derivative $f^{(2)}(t)$ in term of $f(t)$? ...
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1answer
72 views

Radius of convergence of a sum of two series

Assume that the radius of convergence for $\sum_{k=0}^{\infty} c_k x^k$ is $11$ and that the radius of convergence for $\sum_{k=0}^{\infty}d_k x^k$ is $13$. Determine the radius of convergence for ...
2
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1answer
56 views

Bounding a continuously differentiable function using Taylor given the function is bounded by the norm of x

Suppose $0 < r < 1$ and that $f \colon B_1(0) \to \mathbb R$ is continuously differentiable. If there is an $\alpha > 0$ s.t. $|f(x)<\Vert x\Vert^\alpha$ for all $x \in B_r(0)$, prove ...
4
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1answer
102 views

A sort of “Taylor expansion” of a power series

I have the following question. Suppose $$f(x):=\sum_{i=0}^{\infty}c_ix^i$$ is a power series that converges for $|x|<1 + \epsilon$, for some $\epsilon >0$, where $x\in\mathbb{C}$. I can then ...
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1answer
206 views

limit of the error in approximating definite integral with midpoint rule

I want to calculate $\lim_{n \rightarrow \infty} n^2 |\int_{[0,1]}f(x)-I_n(x)|$ where $I_n$ is the integral approximation by midpoint rule: $I_n=\frac{1}{n}\sum_{k=1}^nf(c_k)$ and $c_k$ is the point ...
5
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1answer
191 views

Application for mean value theorem

$f(x)$ is three-times differentiable on $[a,b]$, how to show that there is $\varepsilon\in(a,b)$ such that $$f(b)=f(a)+\cfrac{1}{2}(b-a)[f'(a)+f'(b)]-\cfrac{1}{12}(b-a)^3f'''(\varepsilon)$$
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1answer
56 views

Taylor polynomial

I need your help to solve this question. I tried something, but i can't finish my proof. Let $f(x)$ be a differentiable function in $(0, \infty )$, so that $|f'(x)|$ is bounded there. Prove that ...
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1answer
274 views

Taylor series for logarithm converges towards logarithm

Is there a way to show that the Taylor series around 0 of $f(x) = \ln(1-x)$ converges towards $f$ on the interval $(-1,1)$, just by considering the remainder from the Taylor polynomial? I'm having a ...
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0answers
111 views

using Taylor series to prove an inequality

Prove that if $p^T▽f(x_k)<0$, then $f(x_k+εp)<f(x_k)$ for $ε>0$ sufficiently small. I think we can expand $f(x_k+εp)$ in a Taylor series about the point $x_k$ and look at $f(x_k+εp)-f(x_k)$, ...
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1answer
64 views

Taylor s inequality

Apply Taylor´s inequality to derive the quadratic Taylor approximation of $e^x$ at $x=0$. Could anyone help me out? I tried looking up the definition but I am not sure what is meant by "at $x$ is ...
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1answer
146 views

Bounding approximation error for Taylor polynomial

I've got this problem: Let $f(x) = e^x$. If we aproximate $f(x)$ by $P_4(x)$ in $x_0 = 0$ at $(-r, r)$, find $r \gt 0$ so that the error in the approximation is $\lt 10^{-5}$ What I did is: 1) ...
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1answer
91 views

Exact expansion of functions

Prove that for any twice differentiable function $f: {R}^n \to R$, $f(y) = f(x) + \nabla f(x)^T (y-x)+ \frac{1}{2} (y-x)^T \nabla^2f(z)(y-x) $, for some $z$ on the line segment $[x, y]$. Note that ...
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1answer
186 views

The second order approximation of the Taylor expansion of Characteristic functions:

Let $X$ be a random variable with continuous density $\rho(x)$. Assume that $X$ is symmetric and $\vert X\vert<L$. Since it has a bounded support, all moments of $X$ are well-defined. Let $m_i$ ...
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1answer
321 views

Taylor / Maclaurin series expansion origin. [closed]

Soo we all know Taylor series expansion formula for expansion around expansion point $A(a,f(a))$: $$f(x) \approx \underbrace{f(a)}_{1st~term} + \underbrace{\frac{f'(a)\, (x-a)}{1!}}_{2nd~term}+ ...
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1answer
539 views

Finding the 4th order Taylor expansion of $g(t)= t^3 + 2t^2 + 2t + 1$

Given the function $$g(t) = t^3 + 2t^2 + 2t + 1$$ I would like to find the 4th order expansion of $g(t)$ at $t=t_1$. So far, I have performed the differentiation of $g$, up to $g'''(t)$ w.r.t. $t$, ...
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3answers
892 views

Where do the factorials come from in the taylor series?

Unfortunately, I don't have much detail to give here. But is the general idea to cancel out the constant obtained from taking the derivative. For instance, say my function was ...
0
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1answer
61 views

Expansion of $x^{-1/2}$ at $0$

Regard the function $f(x) = x^{-1/2}$ on the non-negative real line. The point $z=0$ is 'distinguished' because it is the boundary of the domain of $f$, and because $f$ has a pole there. It seems ...
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5answers
123 views

Proof that $2.82<\pi<3.19$

Using taylor expansion of $\cos$ function. What I have is $$1-\frac{x^2}{2}+\cdots-\frac{x^{4n-2}}{(4n-2)!}<\cos(x)<1-\frac{x^2}{2}+\cdots+\frac{x^{4n}}{(4n)!}$$ How would I proceed from here? ...
2
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2answers
84 views

Find $\displaystyle\lim_{x \rightarrow 0} \frac{e^{\sin x} - e^x}{\sin^3 2x}$

I have to find $\displaystyle\lim_{x \rightarrow 0} \frac{e^{\sin x} - e^x}{\sin^3 2x}$ using Taylor polynomials. Here's what I've done so far: $e^x = 1 + x + \frac{1}{2} x^2 + \frac{1}{6} x^3 + ...
1
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2answers
289 views

Calculate the improper integral and the taylor series of $f(x) = \int_{x}^1 \frac{tx}{\sqrt{t^2-x^2}} \,dt$

For the given function $$f(x) = \int_{x}^1 \frac{tx}{\sqrt{t^2-x^2}} \,dt$$ with -1 < x < 1. Calculate the improper integral. Calculate the Taylor series of $f(x)$ at $x=0$ until the third ...
1
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3answers
1k views

How to expand $\tan x$ in Taylor order to $o(x^6)$

I try to expand $\tan x$ in Taylor order to $o(x^6)$, but searching of all 6 derivative in zero (ex. $\tan'(0), \tan''(0)$ and e.t.c.) is very difficult and slow method. Is there another way to ...
2
votes
2answers
512 views

Taylor expansion for $\sqrt{x+2}$

I'm enrolled in Coursera's calculus with a single variable and am trying to solve one of the homework problems. In lecture, it was stated that to expand $\sqrt x$ about $x=a$, you would have: ...