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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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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1answer
61 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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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 ...
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1answer
88 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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1answer
46 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
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1answer
50 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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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 ...
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0answers
21 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 ...
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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 ...
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2answers
53 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 ...
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0answers
29 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 ...
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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.
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1answer
52 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 ...
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0answers
45 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 ...
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2answers
58 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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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 ...
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0answers
27 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} & ...
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4answers
183 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 $, $$ ...
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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] ...
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1answer
48 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?
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1answer
25 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 ...
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3answers
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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 $1$ : $(1+3x)^{1/3}=1+x+o(x)$ ...
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1answer
43 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 ...
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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 ...
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1answer
37 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 ...
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1answer
45 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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2answers
63 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
15 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
60 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) = ...
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1answer
36 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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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 ...
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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, ...
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0answers
25 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 ...
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1answer
48 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: ...
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1answer
29 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 ...
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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 ...
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1answer
42 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 ...
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0answers
184 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) ]) ...
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0answers
22 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 + ...
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0answers
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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 ...
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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
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1answer
47 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 ...
27
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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 ...
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0answers
28 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 ...
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1answer
73 views

What are Bernoulli numbers?

In my calculus class, my teacher said that if one was to try to calculate the maclaurin or taylor series of $\tan x$ by strictly using the definition , then you would run into many problems and your ...
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3answers
77 views

Calculate $\sqrt{5}$ using taylor series to the accuracy of 3 digit after the point $(0.5*10^{-3})$.

I need to calculate $\sqrt{5}$ using taylor series to the accuracy of 3 digit after the point $(0.5*10^{-3})$. I defined : $$f(x)=\sqrt{x}$$ Therefore : $$f'(x)=\frac{1}{2\sqrt{x}}$$ ...
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1answer
26 views

At what points is the function $f(z) = \frac{1}{2+e^z}$ holomorphic?

I need to determine at which points this function is holomorphic. I attempted to use the Cauchy-Riemann equations, but that got too messy and so I'm trying to find another route. In the first part of ...
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3answers
61 views

find taylor series to fourth term

I'm wondering if there is faster method than just calculating derivatives with finding taylor series up to 4 term of function $\displaystyle f(x)=\frac{(1+x^4)}{(1+2x)^3(1-2x)^2}$
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3answers
90 views

How to find the Maclaurin series for $e^{-x^2}$

I don't know how to get $$1-x^2+\frac{x^4}{2!}-\cdots.$$ I think it is too complex, if not impossible, to just use the definition of Maclaurin series. Using the definition: consider the situation ...
2
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1answer
41 views

How do you prove that $\int_{1}^{n}\ln x \,dx \geq \sum_{i = 1}^{n}\ln i -\frac{\ln n }{2}$?

In Upfal's probability textbook Lemma 5.8, he tries to justify $\int_{1}^{n}\ln x \,dx \geq \sum_{i = 1}^{n} \ln i -\frac{\ln n }{2}$ with concavity of $\ln x$, I don't quite follow his argument, can ...