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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Equality of a function and Taylor Series

Does the following function have a Taylor series of the form given below: $$\frac{1}{(1+(\eta z)^n)^p} = ...
4
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0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
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1answer
110 views

taylor series expansion for a rational function

What is the Taylor Series Expansion (function of z ) for where $\eta$, $n$ and $p$ are positive real constants Based on the answers in the comments, does this mean that the taylor series is given ...
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1answer
40 views

Help with Taylor series problem

I am using maple to plot the graphs of e^e^x versus its truncated Taylor series around 0. For small values of x, the two graphs converge nicely, but once x<-3, my Taylor series loses control. Here ...
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0answers
56 views

Error Term of the Taylor series of cosh

I have the Taylor series of cosh $$\sum_{n=0}^\infty \frac {x^{2n}} {(2n)!}$$ and I know that this series converges for all x, but now I want to know if the series represents the function, in other ...
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1answer
121 views

Taylor series of $\frac 1 {1+x^2}$

I have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to construct ...
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1answer
63 views

Should I use Taylor Expansion or the expansion of $e^x$ to express a second order differential (in this case)

I've been given this equation: $(1+x^2)\dfrac{d^2y}{dx^2} + 4x\dfrac{dy}{dx} + 2y = 0$ I've also been told that: $y=1, \dfrac{dy}{dx} = 1$, at $x=-1$ I've been asked to find a series solution of ...
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1answer
59 views

Taylor expansion of polynomial

Intuitively, I would expect the Taylor expansion around $x_0$ of a polynomial in $(x-x_0)$ to be identical to the polynomial. However, I cannot seem to show that/whether this is the case: For a ...
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3answers
54 views

A simple series

I don't do math a long time, so I completely don't remember how to prove that: $$ \sum_{i=1}^\infty \frac{i}{2^i} = 2 $$ Can anybody help me?
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2answers
261 views

Express $(1-z)^{-1}$ as a power series around $z_0=-1+i$.

I need to express $(1-z)^{-1}$ as a power series in powers of $(z+1-i)$. I would like some guidance on the complex analogue of power series and in writing out this particular case. Many thanks for ...
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2answers
54 views

Taylor Polynomial converges to the original function?

If $$P_n(x)=x-\frac{x^2}{2}+\frac{x^3}{3}-..+\frac{x^{2n+1}}{2n+1}$$ (It's taylor series of $\ln(1+x)$ near x=0. Then can I say that: $\lim_{n\to\infty}{P_n(x)}=\ln(1+x)$, please explain why or why ...
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1answer
96 views

Getting the exact value of the Maclaurin series for $\sinh$

I am new to Maclaurin series and I am trying to find the exact value for $\sinh(x)$. Let say I have the highest power of $5$ and value of $x$ is $2$. How do I start?? Sorry I am really new to this.
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0answers
53 views

Multivariate Taylor Polynomial

The Exercise: Calculate the Taylor polynomial of degree 3 of $f(x,y,z)=x^5y^4z^3$ at $(1,1,1)$ in an arbitrary direction $h$. Use Taylor's theorem to get a bound on the remainder when using this ...
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2answers
147 views

Taylor series remainder (Lagrange)

I have this function $f(x) = ln(1+x)$ and I want to come up with the Maclaurin series for it up to $n = 3, a = 0$. I calculate that the remainder $n=3$ is $R_3(x)=-\frac{6}{4!(1+c)^4}x^4, \; c \in ...
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1answer
31 views

$\frac{P(x)}{(ax+b)^n }= \frac{c_1}{ax+b }+\frac{c_2}{(ax+b)^2 }+…+\frac{c_n}{(ax+b)^n }$

Let $n\geq 1$ be an integer, $P(x)$ be a polynomial of degree lower than $n$. Prove, if $a$ is real and $a \neq 0$ then: $$\frac{P(x)}{(ax+b)^n} = \frac{c_1}{ax+b} + \frac{c_2}{(ax+b)^2} + ...
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1answer
70 views

Showing $|a_k | \le 1$

Let $A$ be the closed unit disk $A= \{z \in \mathbb{C}: |z| \le 1 \}$. Suppose $f$ is an entire function whose Taylor series centered at the origin is $$\sum_{k=0}^{\infty} a_kz^k$$ and that $f$ maps ...
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1answer
26 views

Functions and their Taylor polynomials

Given a function $f$ from $\Bbb{R}$ to $\Bbb{R}$, we define $P_{f,n,a}$ to be the Taylor polynomial of $f$ of degree $n$ at $a$ (if the function itself is clear from the context, we simply write the ...
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1answer
44 views
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1answer
40 views

Taylor & MacLaurin series

I have a problem that I can not understand at all. I know how to calculate taylor/macLaurin for $\cos x$, $\sin x$ and $e^x$ etc. However when I have for example: $\sin x \cos x$ or $\sin x + \cos ...
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2answers
48 views

How to prove a series is greater than zero over an interval?

Show that the series $\sum\limits_{k=0}^\infty \frac{(-1)^k(x^{2k+1})}{(2k + 1)!}$ is greater than zero for $0<x\leq \sqrt{6}$ For a function to show something was greater than zero over an ...
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2answers
156 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
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0answers
152 views

Loss of Significance problems - Taylor Expansion

(2) This question addresses the notion of loss of significance. You are encouraged to revisit the Taylor series expansion that you have learned in calculus, as you will need to apply it here. Explain ...
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84 views

Estimate probability of event using moments of a distribution or a Taylor expansion involving the moments

Let's say we have four moments $(\mu_1, \mu_2, \mu_3, \mu_4)$ of a probabilty distribution of a random variable $X$ and the goal is to get the probability $\rm{P}(X \leq t)$ for a certain value of ...
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1answer
68 views

Taylor series about (3+x)/(x+4)*exp(-x) expanded at x = - 4. How do i replicate what I see in wolfram?

I'm puzzled by wolfram alpha's results. If i ask 'series $\frac{3+x}{4+x} (\exp (-x))$ expanded at -4' It will return a series. What method does it use to do this? I'm familiar with Taylor ...
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2answers
90 views

Remainder of Taylor series

The Taylor series of the function $$f(x) = \int_{1}^{\sqrt{x}} \ln(xy)+ y^{3x} dy + e^{2x}$$ at the point $x = 1$ is $$e^2 + (x-1)\left(2e^2+\frac{1}{2}\right) + ...
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0answers
131 views

Convergence of Taylor series of analytic function

Let $f(x)$ be analytic on $D= \{x \in \mathbb R^2: |x|< 1\}$. Then for $x_0 \in D$ there is an open set $U$ such that for all $x \in U$: $\sum_{n=0}^\infty a_n (x-x_0)^n = f(x)$, that is, the ...
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0answers
36 views

Multivariate taylor expansion application

I have to show that, given the operator P such that: $P f(x) = \sum_{|\alpha| \leq m} c_{\alpha} \frac{d^{\alpha} f}{dx^\alpha} = 0$ (Using multivariable multi-index notation) Then the same would ...
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2answers
125 views

Evaluate the sum of the series

Was given the following infinite sum in class as a question* (while we were talking about taylor series expansions of $\ln(1+x)$ and $\arctan(x)$: $$1 + \frac12 - \frac23 + \frac14 + \frac15 - \frac26 ...
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2answers
111 views

Maclaurin polynomial for $\arcsin(x)$

How would I find the 3rd-order Maclaurin polynomial for $f(x) = \arcsin(x)$; with the interval $(0,\frac 3 4)$ to show it in terms of $x$? Would you have to somehow manipulate it to $\dfrac{1}{1+x}$ ...
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1answer
39 views

Taylor expansion for $\frac{\sin(x)}{x}$ at $x=0$?

I don't understand how to expand this function, for example how do I plug $x=0$ into $\frac{\sin(x)}{x}$ ? All derivatives of the function have $x$ in the denominator, I'm just confused. Thanks for ...
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1answer
134 views

Numerical Approximation

we aim to find an approximation of exp(x) a) determine N an integer and g \in (-log(2)/2, log(2)/2] such that x=N*log(2) + g b) Think of a way to use the Taylor expansion with four terms of exp(x) ...
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1answer
91 views

Analytical way of describing centred difference coefficients

I am trying to find an analytical way to describe the finite difference coefficients of various degrees of accuracy of centred difference schemes that approximate the second derivative. For example, a ...
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2answers
61 views

Taylor expansion for Si(x)?

I want to find out what the Taylor expansion of $$F(x) = \int_0^x \frac{\sin(t)}{t} dt .$$ Am I wrong in saying that by the fundamental theorem of calculus, $F'(x) = sin(t)/t$? Should I continue ...
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2answers
122 views

Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$

Prove the following inequality: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ This should be solved using Taylor series. I tried expanding the left to the 5th degree and the right site to ...
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2answers
32 views

Find the Maclaurin series of $f(x)=\frac{x}{x^4+x^2+1}$ [closed]

Maclaurin series for $f(x)$. Thanks.
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2answers
61 views

Maclaurin series for $f(x)=\frac{1}{1+x+x^2} $

What is the Maclaurin expansion of $f(x)=\dfrac{1}{1+x+x^2} $? Thank you! Edit: By multiplying both terms with $ (1-x) $ I got to $\dfrac{1}{1-x^3}-\dfrac{x}{1-x^3}$. Is it correct to transform ...
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2answers
63 views

Solve this limit (Maclaurin or differentiate?)

I have this assignment where I should calculate the limit below: $$ \lim_{x\to0}\frac{\sin 2x}{x\cos x} $$ I can use l'Hospitals rule (because it is a "zero divided by zero"-case) and therefore ...
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3answers
95 views

$\lim_{x\to\infty} \frac{5\cdot5^x+3^x-4^x}{5^x +2^x+27\cdot9^x}$

How can I solve this limit. (Here $x$ belongs to natural numbers $\Bbb{N}$.) $$ \lim_{x\to\infty} \dfrac{5\cdot5^x+3^x-4^x}{5^x +2^x+27\cdot9^x}$$ My try: I tried using L'Hospital, expansions of ...
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3answers
73 views

Taylor expansion - what order would be preferred?

Let say you want to calculate the following limit: $$\mathop {\lim }\limits_{x \to 0} \left( {\frac{1}{{1 - \cos x}}\ln \left( {\frac{{\sin x}}{x}} \right)} \right)$$ Obviously, Taylor Expansion ...
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1answer
383 views

What's wrong in my Taylor series implementation in MATLAB?

I'm trying to code Taylor summation for a function in Matlab, I actually evaluate McLaurin making $x_0=0$, named a in this code after this notation: This is the code I've tried out so far: ...
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1answer
80 views

differentiate arctan (maclaurin?)

I have this assignment: Differentiate this expression: $$ f(x) =\arctan \frac{x-1}{x+1} $$ There is also known that $-1 < x$ (Why is that important?). I do not know how to solve this problem... ...
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5answers
94 views

What does $a$ mean in Taylor series formula?

I'm trying to code the Taylor summation in MATLAB, being Taylor's formula the following: I've also seen $a$ denoted as $x_0$ in distinct bibliography. Problem is that I'm not sure how should I ...
3
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2answers
541 views

Express $\sin nx$ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively

What are the expansions of $\sin nx $ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively? (here $n \in \mathbb N$). Maybe this is solved problem or there is new technique to ...
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1answer
37 views

Tangent and Taylor polynomials

We know that this series $x+ \frac{x^3}{3}+\frac{2x^5}{15}+\ldots$ is convergent in $|x|\lt \pi/2$, furthermore it converges to $\tan(x)$. I would like to know if we restrict to finite terms of this ...
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1answer
87 views

Maclaurin Series Complex Numbers

I'm having trouble getting to the right solution on the function ${z^2\over (1+z)^2}$ ${z^2\over (1+z)^2}$ = ${z^2}$${1\over (1+z)^2}$ = ${z^2}$${1\over (1+z)(1+z)}$ = ${z^2}$${A \over (1+z)}$ + ...
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2answers
117 views

Solving a limit using MacLaurin series

I want to find $$ \lim_{x\to0} \frac{(e^{-x^2}-1)\sin x }{x\ln(1+x^2)}$$ using a Maclaurin series and not using the l'Hôpital's rule. However I can't seem to get it right. Thanks for any possible ...
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1answer
44 views

Asymptotic Expansion in zero of $\frac{1}{\ln(1+x)}$

On wolfram the expansion is: $$\frac {1}{x} + \dfrac{1}{2} ...\,.$$ But I don't understand from where it outside comes the $\frac{1}{2}$ thanks
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0answers
188 views

taylor series of $\sin t/t$

I know that the taylor series expansion around zero of $\sin t/t$ is: $\sum_{k=0}^{\infty} (-1)^k \frac{(t^{2k})}{(2k+1)!}$ , I need to find its radius of convergence. I saw a few solutions that claim ...
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2answers
39 views

Maclaurin Expansion of $\frac{x}{\sqrt{4-2x}}$

Maclaurin Expansion of $\frac{x}{\sqrt{4-2x}}$ up to order 4. I really don't know how to do this, I can't find a helpful Maclaurin Series in my formula book to help me. I want to do $x(4-2x)^{-1/2}$ ...
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0answers
42 views

Landau identification

Calculate taylor series for $x\rightarrow +\infty$ at the higher order allowed by the approximation present in it. $$ \sqrt{x^6+x^5-2x^3+O\left(x^2\right)} $$ I made this: $$ ...