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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Use Taylor Series to arrive at the expression for the forward approximation for a derivative [duplicate]

Use Taylor Series to arrive at the expression for the forward approximation for a derivative. $$f'(x)\approx \frac1h\left(-\frac32f(x)+2f(x+h)-\frac12f(x+2h)\right)$$ I'm not sure how to even go ...
5
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1answer
97 views

Integral of $\int x^{-x} dx$

Question: $\int x^{-x} dx =$ ? Hint: $$ e^{x\ln \frac{1}{x}} = \sum_{n=0}^\infty \frac{x^n}{n!} \left(\ln\left(\frac{1}{x}\right)\right)^n$$ I figure since $\int x^{-x} dx = \int e^{x\ln \frac{1}{x}...
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2answers
80 views

Prove that $\int _ {a}^{b}x^n f(x)dx=0$ implies $f=0$ by using the Taylor theorem

Let $f$ be a smooth function on a closed interval $[a,b]$. Assume that there is $M>0$ such that $\left|f^{(n)}\right|<M$ for all $n\geq1$ and all $x\in[a,b]$. Prove that if $$\int _ {a}^{b}x^n ...
0
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0answers
14 views

How can I show that these 2 Taylor series expansions are equivalent (in 2 dimensions)?

I've been given the following questions: For part $(i)$ I've found that the expansion is given by $f(x+h, y+k)= 1+2h+k+3h^{2}+2hk+\frac{k^{2}}{2}$. However, for part $(ii)$ I've found that $g_{1}(...
0
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1answer
35 views

Why is there a transpose in multivariable Taylor formula? [closed]

Regarding Taylor polynomial of 2nd order, $$ f(x)=f(x_1)+\operatorname{grad}(f(x_1))^T (x - x_1)+\frac12(x - x_1)^T \operatorname{grad}^2f(x_1) (x - x_1) $$ I have the following question: Why is ...
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0answers
38 views

Smoothness of $x\mapsto \frac{1}{1-x}$

I'm reading lecture about taylor expansion but i wonder in following example why he took $x\longmapsto \dfrac{1}{1-x}$ as function in $]-\infty,1[$ of class $\mathcal{C}^{n+1}$ and not in $\mathbb{R}...
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3answers
32 views

$\frac{1}{(1-x)^{2}}=\sum_{k = 0}^{n}(k + 1)x^k+o(x^{n}).$

I would like to show that Taylor expansion of $\dfrac{1}{(1-x)^{2}} $ around $0$ is : $$\dfrac{1}{(1-x)^{2}}=\sum_{k = 0}^{n}(k + 1)x^k+o(x^{n}).$$ My Proof: note that $$\dfrac{1}{1-x}\...
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2answers
24 views

Question about the Taylor theorem and different ways of expressing it

In my lecture notes, it says that the Taylor theorem in one dimension is given by $$f(x+ \delta x) \simeq f(x)+\delta x f_{x} + \frac{1}{2!} (\delta x)^2 f_{xx}+ \dots$$ Conversely, when I was back ...
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1answer
142 views

Taylor series expansion of $e^{-\lambda\tau}$

Taylor series expansion of $e^{-\lambda\tau}$ yields: $e^{-\lambda\tau}=1-\lambda\tau+o(\tau)$ and Taylor expansion of $\lambda\tau e^{-\lambda\tau}$ equals: $\lambda\tau e^{-\lambda\tau}=\lambda\...
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1answer
33 views

Real Analysis: Bounds for derivatives using Taylor's Theorem

Suppose that $f''$ exists on [0,1] and that $f(0)=0=f(1)$. Suppose also that $|f''(x)|\leq K$ for $x\in(0,1)$. Prove that $|f'(1/2)|\leq K/4$ and that $|f'(x)|\leq k/2$ for $x\in(0,1)$. I'm trying to ...
0
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0answers
31 views

Finding a Maclaurin series

I have a question here; suppose $f(x)= x^2\sin(x^3)$ By using the Maclaurin series for sine, find the Maclaurin series for $f$ I understand how to obtain the Maclaurin series for $f$ using the ...
0
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2answers
56 views

Analytic functions equal to all orders in a point are equal on the open interval

Let $A\subset \mathbb{R}$ be open. To make everything clear, my definition of analytic function here is: A function $\psi : A\to \mathbb{R}$ of class $C^\infty$ is said to be analytic if for each $...
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1answer
77 views
1
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1answer
70 views

Asymptotics and little-o notation

I always have issues dealing with asymptotic notation... I am trying to verify the following step: $$\left(1-\frac{t^2}{2n} + o(1/n)\right)^n \to e^{-t^2/2}.$$ To change this into $(1-t^2/(2n))^...
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1answer
40 views

How to show the existence of an entire function

I have been working on this problem for quite sometime. For part (i), I obtained the Taylor series for $4\sin(z) - \sin(4z)$. At $z = -\pi$, the Taylor series is: $4\sum_{n=0}^{n} \frac{(z + \pi)^{...
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2answers
48 views

Compact Form of the Taylor Series

Determine the Taylor Series $\frac{1}{\sqrt{1-x}}$ at $x=0$ I ended up with this: $1 + \frac{1}{2}x+\frac{3}{4}x^2\frac{1}{2!}+\frac{15}{8}x^3\frac{1}{3!}+\frac{105}{16}x^4\frac{1}{4!}$ I am ...
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1answer
86 views

Limitations of fractional derivative approximation with Taylor series

I was playing around with the concept of fraction derivatives, and came across some base functions for which it is defined, namely power and exponential functions $$ \left(\frac{d}{dt}\right)^\alpha ...
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2answers
97 views

taylor expansion of $\sinh(x)$

I would like to find taylor expansion of $sh(x)$ My thoughts indeed, note that : $\sinh(x)=\dfrac{e^{x}-e^{-x}}{2}$ then \begin{align} \sinh(x)&=\frac{e^x-e^{-x}}{2} \\ &=\frac{1}{2}\...
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1answer
60 views

Different convergence radius for different power series of the same function.

i was playing around with $$\frac{1}{x^2+x+1}$$ I got 3 different series's: $$\sum_{n=1}^\infty (\frac{x}{(x+1)^2})^n*(\frac{1}{x}) $$ which converges when $|\frac{x}{(x+1)^2}|<1$ the second ...
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2answers
52 views

To show that the limit of the sequence $\sum\limits_{k=1}^n \frac{n}{n^2+k^2}$ is $\frac{\pi}{4}$

Show that $$\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{n}{n^2+k^2} = \frac{\pi}{4}.$$ I am familiar with Taylor series and Fourier series of the standard functions. I tried to compare with those ...
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2answers
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Taylor series for df

So I understand if I have f(x) under a taylor expansion I can write the terms up to order 2 terms as: f(x)= f(a) + f'(a)(x-a) + [f''(a)*(x-a)^2]/2! +... so I would imagine df(x)/dx = f'(a) + [f''(a)...
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$\ln(1+x)\underset{x\to 0}{=} \sum_{k=1}^{n}(-1)^{k+1}\frac{x^{k}}{k}+o(x^{n})$

I would like to see the setps behind that implication $$ \frac{1}{1+x}\underset{x\to 0}{=} \sum\limits_{k=0}^{n-1}(-1)^{k}x^{k}+o(x^{n-1}) \implies \ln(1+x)\underset{x\to 0}{=} \sum\limits_{k=1}^{n}(...
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1answer
47 views

Leading order Taylor Series Represention of the following function

I am given with this function $$f=\frac{1}{\sqrt{1+af_1(x)+bf_2(x)}},$$ where $$f_1=(1+x^2)^\nu,$$ and $$f_2=x^2(1+x^2)^{\nu-1},$$ where $\nu$ is a rational constant. I would want my $f$ to be of the ...
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2answers
48 views

Taylor expansion for vectors

$$F(x,y)= (x_2-x_1^2) (x_2-2x_1^2)= 2 x_1^4+x_2^2-3x_1^2x_2$$ Where $x^*=[x_1 \ \ x_2]' = [0 \ \ 0]'$ I want to show Taylor expansion of the function for third degree. What I did is that; ...
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1answer
23 views

Reducing terms in the series expansion of a function of two variables

I have a function $f(x, y)$. This function is such that \begin{align} f(0, y)=a\\ f(x, 0)=a, \end{align} where $a$ is a constant. From this, a particular mathematician concludes: Thus if we ...
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1answer
85 views

On the binomial series $(1+\frac{1}{8n})^{1/2}$, where $n$ is an even perfect number

Since $\sqrt{1+8n}=\sqrt{8n}\sqrt{1+\frac{1}{8n}}$, and $\frac{1}{8n}<1$ when $n>1$ is an integer, then we can express the real number $\sqrt{1+\frac{1}{8n}}$ by its binomial series. This series ...
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2answers
54 views

How to prove $\frac{x}{e^x-1}=1-\frac{1}{2}x+\frac{1}{12}x^2+o(x^2),(x\to0)$ using Taylor's Formula?

$$\frac{x}{e^x-1}=1-\frac{1}{2}x+\frac{1}{12}x^2+o(x^2),(x\to0)$$ I have attempted to expand the multinomial $e^x-1$ by using Taylor's Formula, and I got this: $$\frac{x}{e^x-1}=\frac{x}{x+\frac{x^2}{...
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1answer
32 views

approximating $(1-e^{-x})^2$ near $x=0$ with $x^2$ via Taylor expansion

I would like to show that $(1-e^{-x} )^2$ is approximated well near $x=0$ with $x^2$ via Taylor expansion but can't quite seem to complete the job. I know that by expanding the exponential into its ...
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1answer
25 views

How to find the degree- n term in the Maclaurin polynomial of $f(x)=\ln(1+x)$?

How to find the degree- n term in the Maclaurin polynomial of $f(x)=\ln(1+x)$? My Thoughts: The nth term is obviously: $$\frac{f^{(n)}(0)}{n!}x^n$$ But I am stuck here, how do I find the nth ...
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0answers
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Finding the Interval of Convergence for the Sum/Difference of two Power Series

The question: Find the Taylor Series for $$f(x) = \frac{1}{x^2-3x-18}$$ at x = 1. Find the interval of convergence. My work: $$\frac{1}{x^2-3x-18} = \frac{1}{9}(\frac{1}{x-6}-\frac{1}{x+3})...
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1answer
186 views

Cauchy Product of two Taylor Series

I'm probably being a bit stupid here but I've been assigned this question and don't really know where to go with it. Compute the first 5 terms of the Cauchy product of the Taylor Series for $(1-x)^{2/...
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1answer
49 views

Series expansion at infinity

I am trying to find to generalize the limit that involves all rational functions such as $\sum_{n=0}^{l}\frac{{a}_{n}{x}^{n}}{{b}_{n}{x}^{n}}$. I believe the best way of generalizing all of them is ...
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0answers
12 views

Bounding the error for $e^{x+5y}$ taylor polynomial expansion

The exercise asks me to prove: $$|e^{x+5y}-P_1(x,y)|< \frac{3}{2}(x+5y)^2$$ when $x+5y<1$ I don't understand what's the exercise suggesting but I tried this: $e^{x+5y} - P_1(x,y)$ is just ...
5
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1answer
209 views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: $$...
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1answer
30 views

Taylor polynomial of degree 2 of $e^{x^2+x}$

I want to find the Taylor polynomial of degree 2 of $e^{x^2+x}$ and this is what the answer should be: $$e^{x^2+x} = e^{x^2}e^{x} = (1 + x^2 + O(x^4)) (1 + x + \cfrac{x^2}{2} + O(x^3)) = 1 + x + \...
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2answers
79 views

Taylor series of a convolution

The derivation below is from Probability Theory: The Logic Of Science By E. T. Jaynes, chapter 7 "The Central Gaussian, Or Normal, Distribution", p.706 The Landon derivation. Text available online: ...
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1answer
21 views

Taylor expansion of difference of functions

Is the taylor expansion of the difference of functions (more specifically the difference of the same function at different points) simply the difference of the taylor expansions? Since that may be ...
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2answers
40 views

Estimating error using Taylor Polynomial

I have searched and read quite a bit on this subject but I can't get this last bit straight. Reading the other answers did not help me unfortunately for me. Anyway the problem: Suppose I have the ...
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0answers
63 views

Is e^(-1/x) a flat function at x = 0?

Taylor series of e^(-1/x) at x = 0 shows that it is flat function on x= 0. But in every text on flat function I see the example of the function e^(-1/x^2) and not e^(-1/x). I am starting to think that ...
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3answers
46 views

Prove that the series $ \sum_{n=0}^\infty \frac{(-1)^n \cdot x^{2n}}{(2n)!} $ represents $ \cos x $ for all values of $ x $

guys. The question is as stated in the title: prove that the series $ \sum_{n=0}^\infty \frac{(-1)^n \cdot x^{2n}}{(2n)!} $ represents $\cos x $ for all values of $ x $ My doubt is quite theoretical:...
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1answer
23 views

How large need n to be to ensure that Taylor polynomial around x=0 gives a value of sin(pi) which has an error of less than 0.001?

I've found different methods to calculate $n$, but all include that I test it for several $n$. Is it possible to make a general formula that gives me the answer without having to test it, or do I need ...
0
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1answer
23 views

expressing $p , p(p+1) , p(p+1)(p+2)$ as a series

I'm working on arithmetical analysis and more specifically on finite differences. I want to create a series consisting of the following terms : $$f(x_{0} + ph) = f_{0}+ p\Delta f_{0}+ \frac{p(p+1)}{2!...
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1answer
53 views

Taylor Series Expansion of $ f(x) = \sqrt{x} $ around $ a = 4 $

guys. Here's the exercise: find a series representation for the function $ f(x) = \sqrt{x} $ around $ a = 4 $ and find it's radius of convergence. My doubt is on the first part: I can't seem to find ...
0
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1answer
19 views

Write the indicated case of taylor's formula

I have this problem: "Write the indicated case of Taylor's formula for the given function. What is the Lagrange remainder in each case? $f(x) = \ln{x}$ $a = 1, n = 6$ " That's the information I ...
0
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0answers
26 views

Is there a way to separate this function?

let $f(\mathbf r_1,\mathbf r_2) = \frac{1}{|\mathbf r_1 - \mathbf r_2|^2 + a^2}$. Is there a method to represent this as (a series of) separated functions in the form?: $f = \sum \limits _i g_i(\...
0
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0answers
35 views

How to derive analog of power rule for other forms of the derivative?

Introduction We'll be dealing with multiple forms of calculus here. So we'll use $\operatorname{L_d}(f(x))$ to refer to the additive derivative, $\cfrac{df}{dx}$, $\operatorname{P_d}(f(x))$ to refer ...
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2answers
49 views

What is the trick to Taylor expand this function to 4th order?

The function is $ u(x,y)= -x-y-xyu^3$, and I want to Taylor expand $u(x,y)$ around (0,0) in powers of x and y to 4th order. To first order, I differentiated implicitly, and the expansion is: $$u(x,y)...
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0answers
28 views

remainder term error in maclaurin polynomial

Consider function f(x)=$\frac{1}{1-x}$, find the remainder term Rn(Z) of a function of x and n. I now know that $f^{(n)}(x)=\frac{n!}{(1-x)^{n+1}} $ and that $Rn(z)=\frac{f^{n+1}(z)}{(n+1)!}(x)^{n+1}$...
0
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1answer
34 views

Is the remainder of first-order Taylor expansion still continuously differentiable?

Let $f: {\mathbb R}^n \to {\mathbb R}^n$ be a continuously differentiable function. Then, we can rewrite its first-order Taylor expansion at $x \in {\mathbb R}^n$ for $h \in {\mathbb R}^n$ that \begin{...