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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5answers
133 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 ...
0
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1answer
25 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 $arctan(x)$ ...
3
votes
1answer
44 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) ...
0
votes
0answers
23 views

Find the radius of convergence of the Maclaurin series $\ln\left(x^3+\sqrt{x^6+64}\right)$

First you need to expand the function in a Maclaurin series. Then find the radius of convergence of the Maclaurin series. My question: $$f(x)=\ln\left(x^3+\sqrt{x^6+64}\right)$$ My solution: ...
3
votes
1answer
40 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)!}$$ ...
2
votes
2answers
33 views

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.
5
votes
2answers
125 views

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 ...
-3
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1answer
22 views

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.
0
votes
1answer
33 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
29 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.. ...
0
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2answers
18 views

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: $$f(0)=2$$ ...
0
votes
1answer
39 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 ...
3
votes
0answers
36 views

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 ...
0
votes
1answer
77 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 ...
1
vote
1answer
16 views

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, ...
2
votes
1answer
42 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 ...
2
votes
1answer
40 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 ...
0
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0answers
15 views

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 ...
0
votes
1answer
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 ...
4
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2answers
40 views

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 ...
0
votes
0answers
27 views

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 ...
1
vote
1answer
23 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.
0
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1answer
34 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 ...
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0answers
40 views

Develop the Taylor series of $\ln(z^2-5z+6)$ in $z=0$

Also, determine the radius of convergence. $\ln$ is the principal branch of the complex logarithm. What I've tried is splitting the function into $\ln(z-3)+\ln(z-2)$ and then finding the formula for ...
0
votes
2answers
72 views

How to find a Taylor series for $e^{x^2-1}$? [closed]

How do I proceed to write a taylor series expansion for $e^{x^2-1}$? I know the series for $e^x$: it is $1+(x)+(x^2/2!)+\dots$ Edit: Would a Maclaurin series expansion be different?
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2answers
40 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 ...
0
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0answers
30 views

Compute radius of convergence and the first three coefficients of a function

Let $\displaystyle f(z) = \frac{z+1}{(2z+1)(1+ \sin z)}$, with serie expansion $\sum_{n=0} ^\infty a_n z^n$ around zero. Now I want to compute the radius of convergence and the first three ...
0
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0answers
25 views

Taylor expansion of the solutions of the equation $1-4 \cos(\frac{1}{x})+8x \sin(\frac{1}{x})=0$

In following article, I give an example of a function whose derivative at 0 is equal to 1 but which is not increasing near 0. The function is: $$\begin{array}{l|rcl} f : & \mathbb{R} & ...
4
votes
4answers
177 views

The even-numbered coefficients of the Maclaurin series of $ \frac{1}{\cos(x)} $ are odd integers.

Let’s consider $ G(z) \stackrel{\text{df}}{=} \dfrac{1}{\cos(z)} $ as the exponential generating function of the sequence of Euler numbers. How can one prove that in the Maclaurin series of $ G $, $$ ...
0
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1answer
12 views

Reasoning behind method of steepest descent

I am considering the method of steepest descent from my notes. I have written that $$\int_a^b dx e^{g(x)} \sim e^{g(x_0)} \int_{\infty}^{\infty}dx \exp \left[-\frac{1}{2}(x-x_0)^2|g^"(x_0)|\right] ...
0
votes
1answer
22 views

Two variables Taylor's expansion

I guess that Taylor's expansion about $(0,0)$ is useful for finding value of $\dfrac{\partial^{4n}}{\partial x^{2n}\partial y^{2n}} \left (\dfrac{1}{1+x^2+y^2}\right)(0,0) $. How can it do?
1
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1answer
20 views

Estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with 2D Taylor

I need to estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with a Taylor approximation of a two variable function (i.e. x and y). Eventually I managed to pull the (presumably) correct ...
2
votes
2answers
36 views

Find $\lim\limits_{x \to 0} \frac{(1+3x)^{1/3}-\sin(x)-1}{1-\cos(x)}$

I would like to find using Taylor series : $$\lim\limits_{x \to 0} \frac{(1+3x)^{1/3}-\sin(x)-1}{1-\cos(x)}$$ So I compute the taylor series of the terms at the order $1$ : $(1+3x)^{1/3}=1+x+o(x)$ ...
1
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0answers
23 views

Radius of convergence of $x/sinh(x)$

the function $\mathbb{R}\ni x\mapsto \frac{x}{\sinh(x)}\in\mathbb{R}$ can be written in a neighborhood of $0\in\mathbb{R}$ as a Taylor series, i.e. $\frac{x}{\sinh(x)}=\sum\limits_{k=0}^{\infty} a_k ...
1
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1answer
34 views

how to understand Taylor's inequality intuitively?

I am learning the Taylor Series at the moment and I am trying to figure out how to understand Taylor's inequality intuitively. I know you can integrate repeatedly and prove the inequality is ...
0
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3answers
23 views

Power series expansion using Taylors Theorem.

So the function $f(x)=3x^2-6x+5$ needs to be written as a power series expansion around $x=a$ and the goal is to show $x=a$ is $f(x)$ for every $a$. So I started off by finding up to the third ...
2
votes
1answer
30 views

Are the coefficients of a Taylor series bounded when the function is?

Say that I have three real functions $f(x)$, $g(x)$, and $h(x)$ such that $f(x)\le g(x)\le h(x)$ for all real $x$. Additionally, $f(x)$ and $h(x)$ are logarithmically convex. Can I make any definite ...
1
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1answer
31 views

Maclaurin Series - finding the co-efficients for functions that require the product rule

I have just been introduced to the Maclaurin series, and one of the questions I have requires that I find the Maclaurin series for the function $$ f(x) = 3x^2\sin(2x)$$ The way I considered ...
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vote
2answers
56 views

Taylor expansion of $\sin(x)$ and periodicity

Consider that $$\sin x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!}$$ and $$(a+b)^k = \sum_{i=0}^k {k \choose i}a^ib^{k-i}.$$ Then: $$\sin (x + ...
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1answer
12 views

Approximation of monthly payment using Taylor expansion

I am trying to understand what does APR(annual percentage rate) and how it is calculated. Thanks to Wikipedia, I got the formula of monthly payment for a fixed rate multi-year mortgage in the ...
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1answer
52 views

Taylor polynomial for an integral

This is the first time encountering a Taylor expansion along with an integral, so I am wondering how I should proceed. Question: $Consider \space the \space function$ $$F(x) = ...
0
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1answer
20 views

Taylor series expansion approximating an integral?

I need to use the Taylor series expansion of $$\frac{1}{1+3x^2} $$ to find a series approximating $$\int_0^1 \frac{1}{1+3x^2} \, dx $$ and $$\int_0^{1/3} \frac{1}{1+3x^2} \, dx $$ I tried to start the ...
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votes
3answers
50 views

Determine the Taylor expansion for the solution of the differential equation

I'm given the following: $$\begin{cases}\frac{dx}{dt} = t^2x\\ x(0) = 1\end{cases}$$ I'm asked to determine the taylor expansion for the solution to the $t^{10}$ term. $$x(t) = a_0 + a_1 t + a_2 ...
0
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1answer
22 views

Taylor Series Expansion for ${z^2+4z^4+z^6}/(1-z^2)^3$

So I know for sure that the Taylor Series expansion for $1/(1-z^2)^3$ is $\sum {k(k-1)z^{2k-4}/{2}} $ assuming |x|<1. But what do we do with the top? I think its already in the expanded form, ...
0
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0answers
23 views

Find out if $f''(0)$ exist and $x=0$ is a inflection point

Let $f(x)=\begin{cases} \displaystyle\sin(\frac{1}{x})\cdot e^{\frac{-1}{x^2}} &\text{for } x \neq 0\\0 &\text{dla } x=0 \end{cases}$ Find out if $f''(0)$ exist and $x=0$ is a inflection ...
0
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1answer
39 views

Find the radius of convergence of the Maclaurin series $x\cdot ln\left(x^2+\sqrt{x^4+9}\right)$

First you need to expand the function in a Maclaurin series. Then find the radius of convergence of the Maclaurin series. My question: $$x\cdot ln\left(x^2+\sqrt{x^4+9}\right)$$ My solution: ...
1
vote
1answer
14 views

Remainder Taylor series two+ variables

Hard to find an article / tutorial specifically about this subject online.... Can anyone explain / send link to explanation or tutorial regarding how to calculate remainder for multi-variable taylor ...
0
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1answer
24 views

Show that $\sum r^n cos(nx)=rcos(x)-r^2/(1-2rcos(x)+r^2)$

I'm a little unsure about how to approach this. I've been told that we have to use the relationship that $\sum r^n=1/1-r$. However, I'm not too sure what to do with the $\cos(nx)$. Can someone give ...
0
votes
1answer
22 views

Taylor Series expansion and radius of convergence for $e^z+e^{-z}+2cosz/4$

So I did this by taking apart bits of that long equation: $e^z=\sum z^n/n!$, $e^{-z}=\sum(-z)^n/n!$ $2\cos z=e^{iz}+e^{-iz}$ So when we put these together as a Taylor Series, do we just add them ...
1
vote
0answers
171 views

Global and local maxima in a weighted sum of logarithms of linear functionals?

Is is possible to describe, and locate efficiently, the maxima of the function below in the parameters $\mathbf{x}$ $$\sum_{i} p_i \log( N + \sum_j x_j[B_j +(A_j-B_j)\delta_{ij} + min(A_j,B_j) ]) ...