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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Solution to Equation involving Volatility

The following question will have little context, though, it is not relevant. To summarise though, I am trying to find solutions $u$ and $d$ to the following equation given that $d = \frac{1}{u}$: ...
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1answer
17 views

Taylor expansion of second order

I have to find the Taylor expansion of second order of the following functions with center the given point $(x_0, y_0)$. $f(x, y)=(x+y)^2, x_0=0, y_0=0$ $f(x, y)=e^{-x^2-y^2}\cos (xy), x_0=0, ...
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1answer
16 views

Determine Taylor's formula for multivariable equation

Determine Taylor’s formula for the function $f(x, y) = \ln(x+ 2y)$ at the point (1, 0) with remainder term of order three (i.e., the remainder term contains the third total differential). I haven't ...
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1answer
42 views

Maclaurin series for a function

Provided I have the function \begin{equation*} f(x)=(1+x)^{1/x}, \end{equation*} and I want to calculate a 3rd order Maclaurin series, how can that be done without taking direct derivatives (as ...
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1answer
21 views

Power series expansion requirements

Hello stackexchange folks :) I have a question regarding the assumptions made right before you choose to expand or approximate a function by a power series. Specifically I have the function: ...
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Prove $|f'(\frac{1}{2})|\leq \frac{1}{4}$ [duplicate]

Let $f:[0,1]\to \mathbb{R}$ be a function whose second derivative $f'(x)$ is continuous on $[0,1]$. Suppose $f(0)=f(1)=0$ and that $|f''(x)|\leq 1$ for all $x\in [0,1]$. Prove that ...
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How can I show the remainder of this Taylor polynomial $R(h)/h^2$ goes to $0$ as $h$ goes to $0$?

Given the function $f(x, y) = \frac 1{2 - x - y^2}$ I found that the second-degree Taylor polynomial is $$P(x, y) =\frac12 + \frac{x}4 + \frac{x^2}{4} + \frac{y^2}2.$$ How can I show the remainder ...
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73 views

Calculate the Taylor series of $f(x) =\ln( 1 -x +x^2) $ and the domain of convergence

I just stuck at the following exercise: Show that the function f has a Taylor series and calculate it, with $x_0 = 0$. $$ f(x) = \ln{(1-x+x^2)}$$ Because I already know the Taylor series from ...
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A Taylor expansion of $F(x+f(x))$ when $f(x)$ is small

Let's suppose I have a function $F(x)$ and an invertible function $f(x)$. Denote $y=f(x)$ and $u=x+y$. Does the following Taylor expansion (up to two terms) centered at $y=0$ make sense? $$ ...
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19 views

Order $n^{r-1}$ approximation of product given order $(\frac{1}{n^2})$ approximation of terms

I have that $|a_n - (1+\frac{r}n)| \leq \frac c{n^2}$, for $c$ a constant, and am attempting to show that there exist constants $C < \infty$ and $K > 0$ such that the product $b_n = ...
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Determing taylor series from other series

Consider $\cos(x)$ and $\cos(3x^2)$. How to determine the latter's Taylor series from the formers at $a = 0$? I'd write $$\cos{x} = \sum_0^\infty (-1)^n\frac{x^{2n}}{(2n)!}$$ Now, I could just ...
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1answer
22 views

Does $o(|x-a|^n)$ approximation by a polynomial imply existence of derivatives?

While reviewing the topic of Taylor expansion, I've noticed that while in all statements about the $n$th order Taylor polynomial of $f:\mathbb R \to \mathbb R $, it's always assumed that $f\in C^n$, ...
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3answers
80 views

Compute limit using Taylor's expansion

Using Taylor’s expansion, prove that the following limit exists and compute it. $$\lim_{x \to 0}\left(\frac {x^2}{\frac {1}{1-x} - e^x}\right)$$ In this if I am using the taylor series expansion ...
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2answers
31 views

Calculating Lagrange error of a Taylor polynomial approximation

So I am slightly confused when it comes to finding the error of a Taylor series approximation. I know the equation is : $ E_n(x)=\frac M {(n+1)!}(x-a)^{n-1} $ where a is the point that it is ...
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37 views

Two dimension Taylor approximation

Consider function $f:\mathbb{R}^2 \setminus \{(0,0)\}\rightarrow \mathbb{R}$ defined with $f(x,y)=\frac{y-x}{x+y}$. I'm trying to approximate this function on $(0, \epsilon)^{2}$ where $\epsilon ...
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1answer
31 views

Complex number, series representation

Show that for any finite value of $z$ $$e^z=e+e\sum_{n=1}^\infty \frac{(z-1)^n}{n!}$$ For $z=1$ $$f(z)=f(z_0)+\sum f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}$$ equality is checked, but I do not know how to ...
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2answers
29 views

Find the integer $'n'$ for which the given limit is a finite non-zero number.

Find the integer $'n'$ for which the given limit is a finite non-zero number. $$\lim_{x\to 0} \cfrac{\cos^2 x -\cos x -e^x \cos x + e^x - \frac{x^3}{2}}{x^n}$$ I'm almost blind regarding ...
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1answer
22 views

Mc Lauren - Runge Kutta relationship [duplicate]

my question is quite easy(I think). I understand how to apply 4th order Runge Kutta and understand the principle of taylor series (the Mc lauren to be precise) . But I cannot fully understand how the ...
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Easy way to remember Taylor Series for log(1+x)?

Assuming $|x|<1$, if one can easily remember that $$ \dfrac{1}{1-x}=\sum_{n=1}^{\infty}x^{n} $$ then it's easy to mentally derive the following \begin{eqnarray*} \mbox{log}(1-x) & = & ...
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1answer
47 views

Taylor expansion $\ln(1+x+x^2)$ about $x=0$

Is it applicable to use the taylor expansion of $\ln(1+t)$ here and say $t=x+x^2 $ or do I have to take the derivatives?
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Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{ A_{k-1}(t)- A_k(t)}{\alpha+2 t} + \delta_{k \beta} . \tag{1} ...
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Taylor Expansion of Composition of Functions

I have in my lecture notes that $$f(g(x+h))-f(g(x))\approx f'(g(x))\left(g(x+h)-g(x)\right)+f''(g(x))(g(x+h)-g(x))^2$$ He explained can found via taylor expansion, but I try to expand it and am not ...
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48 views

Determine the Taylor Series for $(1+x)^n$ about $x=0$

Having trouble solving this. I get to expanding to this: $$1^n + n(1^{n-1})\cdot\frac {x!}{1!}+n(n-1)\cdot 1^{n-2} \cdot \frac {x^2}{2!} +n(n-1)(n-2)\cdot 1^{n-3}\cdot \frac {x^3}{3!}\dots$$ Where do ...
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1answer
75 views

How do i expand/simplify this quadratic (or quartic?) equation

I'm having trouble doing the following question, was wondering if anyone was able to lend a hand, would be greatly appreciated as i'm not too sure where to start or how to go about this. The ...
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36 views

Taylor Series Formulae

How are the two following forms of the Taylor expansion equivalent? The one I've learnt is $$f(x+h)=f(x)+hf'(x)+\frac{h^2}{2!}f''(x)+...$$ But I've now come across the version $$ ...
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1answer
53 views

Maclaurin Series Approximation of $\sin{x}$

Use first ten terms of the Maclaurin series for $\sin{x}$ to find an approximation to the values of both $\sin{\left(\frac{6\pi}{7}\right)}$ and $\sin{\left(\frac{20\pi}{7}\right)}$? One can say that ...
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2answers
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Taylor polynomial manipulation

Find $\sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{k}$ This is in a section in my book on Taylor polynomials/Taylor series so I assume we have to find some way to manipulate Taylor polynomials to get this. ...
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5answers
148 views

Estimate $\int^1_0 e^{-x^2}\, dx$

Estimate $\int^1_0 e^{-x^2}\, dx$ This is in a section on Taylor series so I would assume that is how it should be solved. I started by using the Taylor series formula for $e^x$ replacing $x$ with ...
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1answer
28 views

Using Taylor series find derivatives of arctan(x)

Using Taylor series for $arctan(x)$, find $f^{(5)}(0)$ and $f^{(6)}(0)$ for $f(x)=arctan(x)$ I figure for this problem I compare the general Taylor series formula to the Taylor series for $arctan(x)$ ...
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1answer
58 views

Are there only a few 'universally convergent' Taylor Series?

The taylor series for $sin(x)$, centered at any point, converges for all x. The taylor series for $e^{x}$ and $cos(x)$ do as well. Thus, taking an algebraic function of these (without division) ...
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Find the value of $a$, $b$ and $c$ for the given limit.

Question - Find the values of $a$, $b$ and $c$ so that $$ \lim_{x\to 0} \cfrac{ae^x - b\cos x +c e^{-x} }{x\sin x} = 2 $$ This is what I've tried yet : For $ x\to 0 $ the numerator must also ...
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1answer
43 views

Can this expression of e be simplified?

Using the maclaurin expansions of coshx and sinhx I came up with $e^x = \sum_{n=0}^\infty$${x^{2n}(2n+1+x)}\over {(2n+1)!}$ Plugging in $x=1$ I got: $$e = \sum_{n=0}^\infty {2(n+1)\over (2n+1)!}$$ ...
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2answers
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Taylor series of $\ln x$ at $x=e$

Like in the title, I need to find taylor series of $\ln (x)$ at $x=e$ I was thinking about changing $\ln (x)$ to $\ln (x-e+e)$ but it lead me to nowhere.
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86 views

Linear Algebra, multiplication of Tensor by vector by vector.

I am deriving some equations and need to know the correct mathematical notation for opening up the brackets of an equation with the following variables: tensor $A \in$ ${\mathbb R}^{l \times l \times ...
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3answers
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General term of Taylor Series of $\sin x$ centered at $\pi/4$

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$? It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$ What power is $(-1)$ supposed to be raised ...
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What is the Taylor series for the function $f(x)=\cos(x)$ centered at $a=(-\pi/4$)? [duplicate]

The title is the extent of the problem. It is a problem from my Calculus II practice test that I am having trouble solving.
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1answer
41 views

Help please with finding the equation and pattern of Taylor Series. (2 problems I have attempted down below).

I didn't want to ask twice so I combined both of my questions together. I have just started on Taylor Series, and I'm not very good at figuring out patterns. First Question Find Taylor Series for ...
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2answers
30 views

How can I make a series expansion of $F(x) = \int_0^x \exp -{(t^2)}\ dt$?

$$F(x) = \int_0^x \exp -{(t^2)}\ dt$$ We need to find the series expansion for $F(x)$. I tried differentiating $F(x)$ but couldn't establish certain pattern so that Taylor series formation may help.. ...
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Find the three non zero terms of the Maclaurin expansion and the radius of convergence of the following function: $f(x)=(4-x)^{1/2}$

Find the first three non zero terms of the Maclaurin expansion and the radius of convergence of the following function: $$f(x)=(4-x)^{1/2}$$ First I found the following to be: $$f(0)=2$$ ...
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45 views

How would I set up the Taylor's Inequality to prove that the function is equal to its Taylor Series expansion?

How would I set up the Taylor's Inequality to prove that the function $f(x) = \frac{1}{x}$ is equal to its Taylor Series expansion centered at $x=1$? I've done the Taylor series expansion, but ...
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Expansion of gamma function

The lecturer wrote down $\Gamma(x-2)=-\frac{1}{2x}+\cdots$ , but I can't figure out where this comes from? It needs to be in this form so that I can cancel the $x$ with the expansion of another ...
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How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
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1answer
46 views

Same Expansion for Different Functions! What's Wrong?

We know, the function $arctan(x)$ is defined as the real number $y$ such that $tan(y)=x$ and $-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}$ From this definition we can derive the well-known ...
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How do I make a Maclaurin series expansion faster?

Suppose I want to approximate to e using the Maclaurin series. In this case, increased accuracy comes with at trade off of computation time. How do I make the Maclaurin series expand faster/ using a ...
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26 views

limit involving a Taylor Polynom

Let $I \subset \mathbb{R}$ be an interval, and let $f: I \to \mathbb{R}$ be a function that's at least n-times differentiable. It needs to be shown that if a polynomial $P(x)$ is of degree $≤ n$, and ...
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1answer
42 views

Taylor series of f(x + a) becomes exponential

In my symmetries of classical mechanics course we have looked at taylor expansions. Our notes claim that; $$ f(x + a) = \sum_{n=0}^\infty \frac{1}{n!} f^{(n)}(x)a^n ≡ \exp{\left( a ...
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2answers
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Maclaurin series expansion for $e^{-1/x^2}$

I am currently extremely confused on how to proceed with the Maclaurin series expansion for my current function. I got my derivatives and I got my formula, however, plugging them in gives me a ...
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Solve a high order polynomial equation in $x$ in the limit $n\rightarrow\infty$

A bit of background. I did a high order WKB theory to calculate the eigenvalues of a potential. The eigenvalues, $E$, are, of course, real since they correspond to a physical problem. My final answer ...
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1answer
24 views

Taylor series help showing expansion

Can someone explain to me why this is wrong, and what I should be doing? I think my method of taking derivatives and pluging in the given value is incorrect.
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1answer
36 views

Taylor series and Maclaurin series problems

Im currently working on these two problems, and Im getting really confused with them. Can someone walk me through them? I will post the work I have so far. http://imgur.com/qXj7zC1 Here is my ...