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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2
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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
145 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
votes
1answer
33 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
44 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 ...
1
vote
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.. ...
0
votes
2answers
38 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
51 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 ...
2
votes
0answers
39 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
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 ...
1
vote
1answer
35 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
48 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 ...
3
votes
1answer
46 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
votes
0answers
20 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
27 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
votes
2answers
47 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
28 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
25 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
votes
1answer
51 views

Taylor series and Maclaurin series problems

I'm currently working on these two problems, and I'm getting really confused with them. Can someone walk me through them? Find the Maclaurin Series for $f(x)=\cos\left(\sqrt x\right)$ and use ...
1
vote
0answers
44 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
75 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?
0
votes
2answers
53 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
votes
0answers
33 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
votes
0answers
26 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
181 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
votes
1answer
19 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
46 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
vote
1answer
23 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 ...
3
votes
3answers
45 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
vote
1answer
40 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
votes
3answers
26 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
36 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
vote
1answer
44 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 ...
1
vote
2answers
62 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 + ...
0
votes
1answer
14 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 ...
1
vote
1answer
58 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
votes
1answer
30 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 ...
-1
votes
3answers
54 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
votes
1answer
23 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
votes
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
47 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
24 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
29 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
35 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
181 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) ]) ...
0
votes
0answers
21 views

multivariable linearization

I have been asked to linearise the fallowing equilibrium points are phy=theta yaw=0 x,y,z=0 The idea I have using V'z as a model: -g+(kcm/m)(cos(phy)cos(thata)*voltages + ...
0
votes
0answers
39 views

How expand an equation in powers of two variables?

Let $$ \varphi=\int\frac{dr}{r^2\sqrt{\frac{1}{b^{2}}-\left(1-\frac{s}{r}\right)\frac{1}{r^{2}}}} $$ Is it possible to expand the above equation in powers of $\frac{s}{r}$?. I know that after ...
-4
votes
1answer
76 views

Find the value of Taylor coefficient $a_5$ for the function $ \int_0^x (e^{-t^2}+\cos t) \, dt$

Find the value for $a_5$ If $ \int_0^x (e^{-t^2}+\cos t) \, dt$ has the power series expansion $\sum_1^\infty a_nx^n$, then find $a_5$ up to three correct decimal places. I think it is a Taylor ...
2
votes
1answer
45 views

Maclaurin Series: Complex Analysis

Question: Use the representation $\sin z = \sum\limits_{n=0}^\infty (-1)^n\frac{z^{2n+1}}{(2n+1)!}$, $|z|<\infty$ to write the Maclaurin series for the function $f(z) = \sin z^2$ and point out how ...
24
votes
0answers
745 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
0
votes
0answers
26 views

Taylor expansion of a power function

I was wondering about Taylor expansions of functions of the form $x^p$, where p is a real number, about $x = 0$. It seems clear how to do it about any other point, but what happens to the series as I ...