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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65 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 : $$f'(x)=\frac{1}{2\sqrt{x}}$$ ...
1
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1answer
34 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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1answer
54 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) = ...
-1
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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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2answers
59 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
13 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 ...
0
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1answer
23 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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556 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 ...
3
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1answer
61 views

Taylor series expansion of the function $f(x)=x \arctan x-0.5 \log(1+x^2)$ about the origin int the region {$|x|\le1$}

Find the Taylor series expansion of the function $\color {green}{f(x)=x \tan^{-1} x-0.5 \log(1+x^2)}$ about the origin int the region {$|x|\le1$} My effort: I know $\displaystyle \log ...
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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
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 ...
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1answer
28 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
60 views

Evaluating $\lim_{n\to\infty}n\sin(2\pi en!)$ [duplicate]

So I need to evaluate $$\lim_{n\to\infty}n\sin(2\pi en!)$$ And yes, I know it was discussed here and similar limit was discussed here, but I didn't feel quite okay with the solutions/hints given in ...
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1answer
46 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
17 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
27 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 ...
4
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2answers
112 views

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
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19 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
30 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 ...
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1answer
75 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
40 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 ...
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0answers
22 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
66 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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1answer
27 views

Is there an interpretation for writing a polynomial in $x$ as a polynomial in $(x-b)$?

Let $Q(x)$ be a polynomial in $x$ of order $n$. The Taylor polynomial of $Q(x)$ of order $n$ developed around $x=b$ (denoted by $P_{n,b}(x)$ ) corresponds to $Q(x)$ written in $(x-b)$. This can be ...
2
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1answer
25 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 ...
2
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3answers
59 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
85 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 ...
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1answer
78 views

How can I solve the integral in the error function $\mbox{erf}(x)$?

To get from this To this series I can't seem find the step-by-step solution anywhere.
2
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1answer
35 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 ...
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0answers
21 views

Multivariate Taylor Series

If $u(x,t)=\alpha + u_1(t)\xi(x,t)+\frac{1}{2}u_2(t)[\xi(x,t)]^2+...$ for small $\xi<0$ apparently $c(u)=c(\alpha)+\xi u_1(t)c'(\alpha)+ O(\xi^2)$ I assume their is some sort of taylor expansion ...
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1answer
65 views

Solve limit of integral through taylor

Show using Taylor expansion that $$\lim_{r\to 0} \frac4{\pi r^2} \int_0^{2\pi} f(a+r\cos \theta , b +r\sin \theta)\cos{2\theta}d\theta = f_{xx} (a,b) - f_{yy}(a,b)$$ where $f:\mathbb R^2 \to ...
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1answer
51 views

Find the Taylor Series expansion of the given analytic function

Find the Taylor Series expansion of the given analytic function $f(z)$, centered at point $z_0$; find the disk of convergence. a) $f(z)=\frac{1}{-2+3i-z}$ $z_0=3$ b) $f(z)=(2-z)\cos{(3z^2)}$ ...
0
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2answers
21 views

taylor series for two variables

The theorem I have been given for this is $$f(x,y)=f(a+u,b+v)=f(a,b)+\sum \limits_{k=1}^{\infty} \frac1{k!} \bigg(u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}\bigg)^kf(a,b)$$ where ...
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1answer
39 views

Find the Taylor series expansion (centered at $z_{0}=0$) of the function $f(z)=\sin(z^3)$.

Find the Taylor series expansion (centered at $z_{0}=0$) of the function $f(z)=\sin(z^3)$. Use this expansion: a) to find $f^{69}(0)$; b) to compute the integral transversed once in the positive ...
3
votes
2answers
29 views

Expand the function in a Maclaurin series $\ln(5\cos^{3}(x))$

$$\ln(5\cos^{3}(x))$$ Need to be expanded: $x^{4}$ I need to end this problem. So I laid the beginning of the function. $$\cos x=1-\frac{x^2}{2!}-\frac{x^4}{4!}+o(x^4)$$ ...
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1answer
29 views

Taylor/ Maclaurin Series: Solving for x

Hi guys I was wondering how to do this question. I'm not sure what method to use.
2
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1answer
52 views

Reverse engineering a Taylor expansion 2

So there is the sum: $$S(x) = \frac{x^3}{3(1!)} + \frac{x^6}{6(2!)} + \frac{x^9}{9(3!)} \text{ }...$$ and we are instructed to find the sum of the series in a small expression. I took the derivative ...
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0answers
32 views

Showing a hyperbola in polar form approaches two asymptotes

Consider a curve given in polar coordinates by $r(θ)=\frac{1}{1+e\cos\theta}$, where $e≥0$. When $e>1$, show that the curve approaches two asymptotes, find them and sketch the curve. Hint: If the ...
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0answers
21 views

Taylors formula using little-o notation proof argument (continuity)

Im trying to prove the following: Let $f: I \to \mathbb{R}$ be $C^n$ on $I \subset \mathbb{R}$ and $P_n$ be the $n$'th degree Taylor polynomial with $a$ as the expansion point then $$ f(x) = P_n(x) ...
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15 views

Taylor series of Lagrangian

Take a look at the Lagrangian defined here. $L=\frac12 a(q)\dot q^2 - V(q)$. You can think of $a$ and $V$ as functions. It seems as though $L$ depends only on $q$. If $q_0$ is a point for which ...
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0answers
44 views

Finding a Taylor Expansion for the following:

I have: $$\frac{1}{1-z}$$ for $z_0=i$. I have no idea how to do the Taylor Series expansion of this, around $z_0=i$, and then show it summation form. I have: $\frac{1}{1-z} = ...
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1answer
61 views

Find the Taylor expansion of $\frac{1}{x^2+2x-3}$ around $x=-1$.

Find the Taylor expansion of $\frac{1}{x^2+2x-3}$ around $x=-1$. What is its radius of convergence? So I write the fraction as $\frac{1}{(x-1)(x+3)}$ and what should I do now?
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1answer
14 views

Taylor Expansion of Inverse of Difference of Vectors

I am trying to derive the multipole moment of a gravitational potential, but I'm getting stuck on some math I believe. So basically the problem is finding the Taylor Expansion for ...
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1answer
61 views

Taylor series of mandelbrot bulb boundaries

What I am looking for is a way to find an approximation to the boundaries of hyperbolic components of the Mandelbrot set. I would like to be able to write a program to find the equations which ...
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0answers
195 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
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2answers
44 views

Taylor series Expansion

I'm a little confused as to what they are asking. all the examples of taylor series expansion I have seen use x instead and I'm not sure how I would go expanding these series.
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0answers
24 views

Taylor series of $(1-x)^b$ $_2F_1(a,b;c;x)$: when to stop?

Let $f(x)= (1-x)^b$ $_2F_1(a,b;c;x)$, where $0<x<1$ and $a=(K-1)d$, $b=K$, c=$Kd$ (with $a$, $b$ and $c$ are positive and $K>d$ ). I need to derive the Taylor series of the corresponding ...
1
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1answer
44 views

The Taylor coefficients of a function of the form $\exp\circ f$, where $f$ is a power series

Let $(a_1, a_2, \dots) \in \mathbb{R}^\infty$ be a fixed sequence of real constants, and suppose the rule $$ x \mapsto \sum_{n = 1}^\infty a_n x^n $$ defines a function from the nonempty open interval ...
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0answers
24 views

General form for series coefficient of Taylor series expansion of $(x+1)^{1/x}$

What is the general form for the series coefficients of Taylor expansion of $(x+1)^{1/x}$? The first few terms are as follows: $$e-\frac{e x}{2}+\frac{11 e x^2}{24}-\frac{7 e x^3}{16}+\frac{2447 e ...
0
votes
3answers
41 views

Taylor Series for $\frac{1}{1+e^z}$ and radius of convergence

I have done some manipulation and got that $$\frac{1}{1+e^z} = \sum_{n=0}^\infty \frac{n!}{n!+z^n}$$ by the fact that: $$\frac{1}{1+e^z}= ...