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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105 views

How to use Taylor series to get $e^x\geq1+x$

I know that from $$e^x=\sum_{i=0}^\infty \left(\frac{x^i}{i!}\right)$$ we can get the inequality $e^x\ge1+x$. But how?
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2answers
54 views

How do I evaluate this without using taylor expansion :$\lim_{x \to \infty}x^2\log(\frac {x+1}{x})-x\ $?

How do I evaluate this without using Taylor expansion? $$\lim_{x \to \infty}x^2\log\left(\frac {x+1}{x}\right)-x$$ Note: I used Taylor expansion at $z=0$ and I have got $\frac{-1}{2}$ Thank ...
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0answers
41 views

Radius of convergence of a Taylor-series

I came across the following question. Let $c \in \mathbb{R}$ and let $f: \mathbb{R} \to \mathbb{R}$ be defined by: $$ f(x) = \frac{1+ c x^2}{1+ x^2}$$ Let $c \neq 1$. Determine for $a=0$ the ...
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0answers
17 views

Functions on a connected subspace

For one of my courses I came across the following question. Let $U$ be a connected supspace of $\mathbb{C}$ and let $f: U \to \mathbb{C}$ and $g: U \to \mathbb{C}$ be complex analytic (eg complex ...
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0answers
23 views

Can I have a proof of the Taylor Series of $f$?

My textbook states the following: Taylor Expansion: $f(x)=\frac{f(a)}{0!}+\frac{f^{\prime}(a)(x-a)}{1!}+\frac{f^{\prime\prime}(a)(x-a)^2}{2!}...$ I actually have used this a lot in my own personal ...
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0answers
83 views

How to prove the following Taylor expansion for twice differentiable functions

I want to prove the following: Let $f(x)$ be a twice differentiable function. Then, $$\begin{array}{l} \exists t \in \left[0,1 \right ] \; s.t., \\ f\left(y \right ) = f\left(x \right ) + \left(y-x \...
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2answers
24 views

Evaluate limit of function through its series expansion

The MacLaurin series expansion of the function $$f(x) = \frac{1}{1 - x}$$ is $$f(x) = 1 + x^2 + x^3 + x^4 + \ldots = \sum_{i = 0}^{+\infty} x^i, \ (x \neq 0)$$ so all the powers of $x$ taken with ...
2
votes
1answer
195 views

Mistake on a Major Maths Website

I think I have found two massive errors on Math.com but I throw my logic out here for a third party to verify as I've been doing Contour Integrals for about 10 hours straight now so I am very tired. ...
4
votes
0answers
74 views

Asymptotic expansion for the solution of linear KDV eq.

Hi, The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the ...
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0answers
22 views

Differential equation with expansion

In this problem of physical origin (Newton's equation of motion of a charged particle in an electric field), can you help me with equation (1.7)? I get more or less where it comes from, but I could ...
2
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1answer
41 views

Surprising pattern of big primes in a derivative series

A pattern of large prime numbers appeared where I was not expecting one, and I'm fascinated to know why. I needed to calculate the Taylor Series for $$f(x)\equiv\ln\left(e^{(x+1)^{2}}+e^{(x-1)^{2}}\...
2
votes
3answers
51 views

Approximation of non-analytic function

I have a function which is of the form \begin{equation} f(x) = \frac{1 - x^{1/2} + x - x^{3/2} + \ldots}{1+x^{1/2} - x + x^{3/2} - \ldots}. \end{equation} Intuitively, I would assume that for small $x$...
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0answers
85 views

Series solution of an ODE (taylor series, infinite series)

We have the ODE: $$y' = -3y$$ Assume that the solution of the ODE can be written as a Taylor series in the form: $$y(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n + \ldots$$ Using this infinite ...
4
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6answers
144 views

$\lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $ solution?

I recently took an math exam where I had this limit to solve $$ \lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $$ and I tought I did it right, since I proceeded like this: 1st I ...
4
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1answer
27 views

If $0<|z|<1$, show that $\frac{1}{4}|z|<|1-e^z|<\frac{7}{4}|z|$

My question: If $0<|z|<1$, show that $\frac{1}{4}|z|<|1-e^z|<\frac{7}{4}|z|$ ($z$ is complex) what I have tried: I tried to expand the middle term in its Taylor series but I can't get ...
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2answers
43 views

Finding the explicit sum of $ \sum_{n=1}^\infty n(n+1)(\frac{1}{2})^n $

I need to find the sum of the following series as an explicit formula: $ \sum_{n=1}^\infty n(n+1)(\frac{1}{2})^n $. I can't express it as a geometric series $ \sum_{n=1}^\infty ar^{n-1} = \frac{a}{1-r}...
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1answer
51 views

Convergence radius of $\sum_{n=1}^\infty (-1)^nn^nx^{n^2} $

I need to find the convergence radius of the series $$\sum_{n=1}^\infty (-1)^nn^nx^{n^2} $$ I have tried using the ratio test $$ \lim\limits_{n \to \infty} \lvert \frac{a_{n+1}}{a_{n}}\rvert $$ and i ...
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0answers
34 views

How can we show that $\sum\limits_{n=1}^\infty |c_n| < \infty$ when $c_n$ is taylor series coefficients of $f(z): \mathbb{C} \rightarrow \mathbb{C}$?

I'm trying to understand a solution to old exam question. And i have trouble understanding and verifying (for myself) one of the steps. if $f(z): \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic on ...
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1answer
35 views

Evaluate the following limit using Taylor series.

What is the limit, when $x\to0$, of $$\frac{4\tan x - 4x -\frac{4}{3}x^3}{x^5}?$$ I'm not sure how to expand this using the Taylor series.
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1answer
55 views

An error bound for a finite difference approximation to the second derivative

I want to show that $$|\delta_{h,r}f(x)- f''(x)| \leq \frac{11}{12} h^2 \|f^{(4)}\|_{\infty}$$ where $$\delta_{h,r}f(x)=\frac{1}{h^2} (2f(x)-5f(x+h)+4f(x+2h)-f(x+3h))$$ I applied the Taylor expanson ...
0
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1answer
31 views

Find limit using Maclaurin series (remember the importance of big O notation)

I have a problem that sounds like this: Find the limit $$\lim_{x\rightarrow 0} \frac{14\tan(6x)-84x}{6x^3}$$ using Maclaurin series, and don't forget the importance of big O notation. I have tried ...
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1answer
34 views

How to find the best value of K ? (Big-O)

Can anyone help me with finding out the best value of K in the following assertion as x goes to 0: $$cos(x)-1+x^2/2=O (x^k)$$ Thanks in advance! Ali
2
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2answers
118 views

Convert Power Series to function

I tried to solve the attached Power Series, however I can't get to the right answer. I wrote down the correct answer at the top-right of the page. Appreciate your help!
0
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1answer
47 views

How to expand $x/\sin x$?

Please help me, I have no idea how to solve this. I know the expansion of $\sin x$ but am not sure if it will apply for $1/\sin x$.
0
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2answers
101 views

How to fastest approximate definite integrals

I know that a definite integral is a limit of Riemann sums. So if one wanted to estimate a definite integral (because one might not be able to find an antiderivative), then one can just take enough ...
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1answer
57 views

Number of terms of $\sin(x)$ required for maximum error of less than $10^{-7}$

Neglecting round-off error, how many terms of the Maclaurin series for $\sin x$ are required to obtain a maximum error of less than $10^{-7}$ in the range $[0,\pi]$? This is part (b); in part (a), I ...
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1answer
35 views

What is the expansion?

I encounter the following formula in some textbook. However, I can not understand what the expansion in this formula is. Is there anyone giving some tips? $$ \begin{align} \alpha &= -\frac{iU}{2 \...
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1answer
39 views

Help me show that there exists an $x\in (\pi/2,\pi)$ such that $\sin x=\frac{x}{2}$

Question Show that there exists an $x\in (\pi/2,\pi)$ such that $\sin x=\frac{x}{2}$ My Solution The way I did it is: $$\sin x=\frac{x}{2}$$ $$x=2\sin x$$ Using Taylor Series $$x=2\left(x-\...
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2answers
69 views

How is Taylor expansion a generalization of linear approximation? [closed]

The concept of derivative is related to linear approximation of a function: $$f(x)\approx f(a)+f'(a)(x-a)$$ I was told that this linear approximation is generalized by the Taylor series. What does ...
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0answers
22 views

$\lim_{x\rightarrow x_0 }\frac{f^{(k)}(x_0)x^k + f^{(k+1)}(c)(x-x_0)^{k+1}}{g^{(k)}(x_0)x^k + g^{(k+1)}(c)(x-x_0)^{k+1}}$

Doing a division $\frac{f(x)}{g(x)}$ of two Taylor polynomials of functions $f$ and $g$ of the form $$f(x) = T_{n,x_0}f(x)+R_{n,x_0}f(x) = \frac{f^{(k)}(x_0)}{k!}x^k + \frac{f^{(k+1)}(c)}{(k+1)!}(x-...
1
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3answers
70 views

Taylor series of arctan(x) (Spivak)

At p. 388 of Calculus, Spivak gives a formula: $$\frac{1}{1+t^2} = 1 - t^2 + t^4 - ... + (-1)^nt^{2n} + \frac{(-1)^{n+1}t^{2n+2}}{1+t^2}$$ Which can be integrated to find $\arctan(x)$. I don't ...
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1answer
19 views

Big-O division in $\frac{f(x)}{g(x)}$ of Taylor polynomials

Here p.4 bottom is a proof that proves a property of $\frac{f(x)}{g(x)}$, the quotient of two Taylor polynomials of f and g. For two Taylor polynomials of $f(x) = \frac{f^{(k)}(x_0)x^k}{k!}+o(x^k)$ ...
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1answer
36 views
0
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0answers
33 views

Question on the deduction of the integral form of remainder of Taylor series

[Theorem deduction] First we have a generalized form of integration by parts $$ \begin{align} \int u(x)v^{(n+1)}(x)dx=\sum_{i=0}^n(-1)^iu^{(i)}v^{(n-i)}+(-1)^{n+1}\int u^{(n+1)}(x)v(x)dx &&(G) ...
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0answers
23 views

Domain specification of derivative extension.

Given the definition of Taylor expansion: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$ We can find the $m$'th derivative of $f(x)$ quite easily: $$\frac{d^m}{dx^m} f(x) = \sum_{n=0}^{\...
0
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2answers
42 views

A particular Taylor Expansion

As we know, the Taylor Expansion we usually see is \begin{equation} e^W = \sum_{n=0}^{\infty}\frac{W^n}{n!} \end{equation} but today I see another equation: \begin{equation} e^{-W}+We^{-W} = \...
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0answers
25 views

How to write the symmetric Hessian matrix for a log function?

Say f(x,y,z) = $y*ln(cos(z)+x^2)$ How would I write this as a Hessian matrix? Would this be the right step I need to take in order to calculate the second-order Taylor polynomial for the function?
2
votes
1answer
150 views

Estimating error in binomial series

I am having trouble with estimating the error in this exercise: Estimate the value of the following integral with error less than 0.0001 $$ \int_{0}^{0.2}\sqrt{1+x^3} \, dx $$ So here im using ...
2
votes
2answers
24 views

Taylor polynomial of order $n$ for a polynomial of degree $n$

I noticed that the Taylor polynomial of order $n$ for a polynomial function of degree $n$ is identical to the function. I tried to understand the reason but couldn't really figure it out. Any input on ...
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3answers
51 views

How do you write the taylor series for (arctan(x))^2

I know that the taylor series of $\arctan{(x)}$ is $x - \frac{x^3}3 + \frac{x^5}5 + \dotsc$ In order to square it I would have to multiply it by itself. $(x - \frac{x^3}3 + \frac{x^5}5 + \dotsc) \...
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1answer
27 views

Taylor expansion with geometric series [closed]

Expand $1/x $ a taylor series around $x=1$. Use $\frac {1}{1-(1-x)}$ and used the geometric series. I know $\sum (1-x)^n = \frac {1}{1-(1-x)}$, $-1 <x -1<1$
2
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1answer
113 views

Finding an upper bound on the Taylor remainder

Any help with the proof I have posted for the following question is greatly appreciated; I have listed my particular issue at the end. Thank you! Let $p \geq 0$. For $n=1,2, \dots,$ find $P_n(x)$ ...
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0answers
57 views

If $0< 2\varepsilon < \sigma^2 < 1$ then $\prod\limits_{i = 1}^n (1 + \varepsilon + \sigma \xi _i )$ converges almost surely to $0$

I posted this question a few days ago and there were some errors in my post. I have fixed them and it should be all right now. Hope someone can help with my confusion. Let $(X_n)_{n\ge0}$ denote an ...
0
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1answer
43 views

Why can't I get good approximation when choosing values away from point of expansion? (Taylor series)

I was in the middle of doing a computing project assigned to me when I came across the question. $\operatorname{P}_N(x)$ is the taylor polynomial for $f(x)=\ln(x)$ expanded around pouint $x_0=1$ $$\...
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0answers
39 views

Find the cubic Maclaurin polynomial for $\sinh x$ and estimate the remainder on the interval $|x| \le 1 $

I'm working through the above problem, and am having trouble deciphering what it means by "estimate the remainder on the interval $|x| \le 1$". I found the Maclaurin polynomial to be $$P_3(x) = x + \...
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1answer
32 views

Taylor's Theorem for 2D function

I am doing a numerical analysis course and we are looking at numerical methods for solving initial value problems. For example: methods such as $y_{n+1} = y_n+\frac{h}{2}(f(t_n,y_n) + f(t_{n+1}, y_{n+...
1
vote
1answer
39 views

Proofs for Taylors theorem and other forms

Let $f \in C^k[a,b]$.Show that for $x,x_0 \in [a,b]$, $$f(x)=\sum\limits_{j=0}^\mathbb{k-1}{{1\over j!}f^{(j)}(x_0)(x-x_0)^j}+{1\over k!}{\int_{x_0}^x f^{(k)}(t)(x-t)^k \,dt}$$ and after this use this ...
4
votes
0answers
178 views

Proof that $\oint_r d(x,N + n) < 0 $?

Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$. And also $0 < D^M f(0) < D^{M-1} f(0)$. Let $0<T<1$ and $n$ a ...
1
vote
1answer
36 views

how to find taylor serie for 1/z with |z| > 0?

I have the following and I need to give the Laurent development for |z| > 0. The Laurent development in this form : and to give few a(n) coefficients How can it be done? normally we use the ...
0
votes
2answers
30 views

Taylor series, identify radius of convergence

I have the following function : I need to find it's radius of convergence with z0 = 0. The function is analytic everywhere except where 1 + sin(iz) = 0 (to my ...