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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higher order asymptotic expansion for likelihood ratio

I have been studying Hayakawa(1975) and (1977) and was wondering if anyone has already computed higher order terms for his expansions following his framework. I'd be very happy if someone could ...
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19 views

Show that $\tfrac{(-1)^{n}}{\cos(n)+n^{\tfrac{3}{4}}}=\tfrac{(-1)^{n}}{n^{\tfrac{3}{4}}}+\mathcal{O}\left( \tfrac{1}{n^{\tfrac{3}{2}}}\right) $

I would like to show that : $$\dfrac{(-1)^{n}}{\cos(n)+n^{\tfrac{3}{4}}}=\dfrac{(-1)^{n}}{n^{\tfrac{3}{4}}}+\mathcal{O}\left( \dfrac{1}{n^{\tfrac{3}{2}}}\right) $$ by starting from the left side ...
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2answers
48 views

How would Taylor Series work?

I wish to calculate sine of any given an angle without using the functions that come with programming language and devices. I have written a small code in Python which can be found here. Using the ...
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1answer
61 views

Taylor expansion $f(x)=f(0)$

The following taylor expansion of the function $f(x)$, requires $f(x)$ to have a derivative up to what order? $$ f(x)=f(0)+f'(0)x+f''(0)x^2/2+\mathcal{O}(x^3)$$ My solution: Based on the Taylor'...
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48 views

Show that $(-1)^{n}\sqrt[n]{n}\sin(\frac{1}{n})=\tfrac{(-1)^{n}}{n}+\mathcal{O}\left(\tfrac{\ln(n)}{n^{2}} \right)$

I would like to show that : $$(-1)^{n}\sqrt[n]{n}\sin(\frac{1}{n})=\dfrac{(-1)^{n}}{n}+\mathcal{O}\left(\dfrac{\ln(n)}{n^{2}} \right)$$ by starting from the left side and get the right side : My ...
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3answers
64 views

Show that $(-1)^{n}\left(\sqrt{n+1}-\sqrt{n} \right)=\tfrac{(-1)^{n}}{2\sqrt{n}}+\mathcal{O}\left(\tfrac{1}{n^{\frac{3}{2}}} \right)$

I would like to show that : $$\fbox{$(-1)^{n}\left(\sqrt{n+1}-\sqrt{n} \right)=\dfrac{(-1)^{n}}{2\sqrt{n}}+\mathcal{O}\left(\dfrac{1}{n^{\dfrac{3}{2}}} \right)$}$$ by starting from the left side ...
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2answers
39 views

Taylor polynomial in function composition

I have the Taylor polynomial of a function f(x): $$4-5x+2x^2$$ and the Taylor polynomial of a function g(x): $$2+\frac{1}{2}x-\frac{1}{8}x^2$$ Both about $$ x=0$$ How can I calculate the Taylor ...
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1answer
26 views

Taylor expansion of Crystal Field potentials

I am trying to work through Michael Tinkham's "Group Theory and Quantum Mechanics". In discussing crystal field theory he uses the following example: We start with an atom at the origin. We want to ...
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2answers
57 views

Show that $\dfrac{(-1)^n}{n+(-1)^n\sqrt{n+1}}=\dfrac{(-1)^n}{n}+\mathcal{O}\left(\dfrac{1}{n^{\frac{3}{2}}}\right)$

How can i prove that $$\frac{(-1)^n}{n+(-1)^n\sqrt{n+1}}=\frac{(-1)^n}{n} +\mathcal{O}\left(\dfrac{1}{n^{\frac{3}{2}}}\right)\tag{$*$}$$ using the following method : note that : $(1+x)^{\...
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3answers
67 views

Convergence of the series $\sum \frac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}}$

To prove that nature of the following series : $$\sum \dfrac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}}$$ they use in solution manual : My questions: I don't know how to achieve ( * ) ...
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0answers
29 views

Difference/switch between big/small o in taylor series

for example i only know taylor series with small o is there anyway to switch from small o to big o in taylor series and why when we want to see the nature of some series we use taylor series with ...
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1answer
65 views

Can the series $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} x^{n}$ be summed? [closed]

Can the following series $$\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} x^{n}$$ be summed?
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2answers
28 views

Find the Taylor-series expansion of a square of a rational function of a complex variable

I've been trying to find the Taylor-series expansion of the following function: $$ f(z)=\left ( \frac{1+z}{1-z} \right )^2 $$ az the origin : Z0 = 0. also I would like to find the region of ...
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0answers
29 views

Taylor polynomial reminder

Page 277 of Apostol's Calculus 1 has the following theorem: Let $P_{n}$ be a polynomial of degree $n\geq1$. Let f and g be two functions with derivatives of order n and assume that $f(x)=P_{n}+ x^ng(x)...
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1answer
41 views

Does the Taylor series of $e^{f(x)}$ converge everywhere?

In STAT 110, the professor says "the Taylor series of $e^x$ converges everywhere, and then proceeds to convert: $${e}^{t^2/2} = \sum_{i=0}^\infty \frac{{(t^2/2)}^n}{n!}$$ I understand that the ...
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1answer
48 views

Finding the 11th term of $\frac{7x^2+2x+6}{(x+2)(x^2+1)}$ Taylor expansion

Given $f(x)=\frac{7x^2+2x+6}{(x+2)(x^2+1)}$, find $f^{(11)}(0).$ I understood that we first need to use partial fractions to simplify the function. $$\frac{7x^2+2x+6}{(x+2)(x^2+1)}=\frac{A}{(x+2)}+\...
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2answers
34 views

Using substitution while using taylor expansion

I am trying to prove to myself why can substitution be used while using taylor expansion. for example: the taylor expansion of $e^{(x-3)^2}$ around $a=3$ and order of $6$ can be done by first ...
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1answer
45 views

Derivative of definite integral using Taylor's theorem

So I want to obtain the following form: For $x, p \in R^n$ $$\nabla f(x+p) = \nabla f(x) + \int^1_0 \nabla^2 f(x+tp)p \,dt$$ for $f$ twice continuously differentiable and $t \in (0,1)$. Taylor's ...
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2answers
65 views

radius of convergence of $1/(1+z^2)$ about $z=2$ using geometric series approach

I would like to calculate the radius of convergence of $f(z)= 1/(1+z^2)$ about $z=2$ using the geometric series approach. Let me first state that according to a theorem, the radius of convergence ...
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1answer
42 views

Does the pointwise convergence of a sequence of Taylor series imply the convergence of the coefficients?

For each fixed $n\in\mathbb N$, let $(a_k^{(n)})_{k\in\mathbb N\cup\{0\}}$ be a real sequence and let $(a_k)_{k\in\mathbb N\cup\{0\}}$ be another real sequence. Suppose the following: there exists ...
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2answers
40 views

Approximating Functions with Polynomials in Taylor Series

I'm having difficulty with a series of problems I've been working on and I can't seem to find a straightforward explanation of how to solve them. The problem is approximating functions. $f(x)=x^{1/3}$...
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3answers
483 views

Faster way to find Taylor series

I'm trying to figure out if there is a better way to teach the following Taylor series problem. I can do the problem myself, but my solution doesn't seem very nice! Let's say I want to find the ...
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2answers
44 views

What is $e^{A}$ where A is an anti-diagonal matrix

I am trying to get a closed form for the matrix produced by the following operation: $$e^A$$ where $A$ is an anti diagonal matrix, say, of size $2\times 2$: $$A=\begin{pmatrix} 0 &b \\ c &0 \...
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1answer
25 views

Finding Taylor series without using derivatives

If $\displaystyle f(z) = \frac{e^{iz}}{z^2-1}$ then we can set $g(z)=e^{iz}$ and $h(z)=z^2-1$. The Maclaurin expansion for $e^{iz}$ is $$\sum\limits_{n=0}^\infty \frac{(iz)^n}{n!}$$ so $\displaystyle ...
4
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2answers
231 views

Sum and product of analytic functions that is not analytic

The function $$f(x) = \frac{2 + \cos x}{3} (2π - x) + \sin x$$ is the sum/product of analytic functions ($\cos(x)$,$\sin(x)$, linear), but all it's derivatives at $2\pi$ are $0$ ($f^n(2\pi)=0$). I ...
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2answers
48 views

How to use the generalized binomial theorem to produce the power series of $(1-x)^{1/2}$ [duplicate]

I am trying to see how to get from $\sqrt{1-x}$ to the power series $\displaystyle\sum_{m=0}^\infty\frac{-1}{2m-1}\,{2m \choose m}\,\frac{x^m}{4^m}$, ideally using the generalized binomial theorem. I ...
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Asymptotic Expansions of a Generalized Hyper-Geometric Function

Let $t>0,x>0$, and $$\{a_1,a_2,a_3\}=\{2, 2, 9/8 - (i t)/2\}$$ $$\{b_1,b_2,b_3,b_4\}=\{1, 1, 3/2, 17/8 - (i t)/2\}$$ We are looking for the asymptotic expansions of a generalized hyper-...
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1answer
40 views

Finding Laurent and Taylor series

I need to find both a Laurent and a Taylor expansion. $$f(z)=\frac{z}{(z-1)(z-2)} = \frac{-1}{(z-1)}+\frac{2}{(z-2)}$$ If I choose $z_0=0$ $$f(z)=\frac{1}{(1 + z)} - \frac{4}{\left(1 - \frac{z}{4}...
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4answers
47 views

Taylor series of function

I need to find the Taylor series of $f(x)=x^2c^x$ centered at $x_0=0$. I tried to take the first three derivatives to look for a pattern, but apparently, this is not the best way, the derivatives ...
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0answers
21 views

Quadratic taylor polynomial of $f(x, y) = x^4 + x^2y^2 + y^4$

Given $$f(x, y) = x^4 + x^2y^2 + y^4,$$ I would like to calculate the taylor polynomial of degree $2$ at $(0, 0).$ Approach This exercise confuses me. I simply tried to follow the ...
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1answer
40 views

Use taylor expansion to study graph geometrically

I have a line which increases very very quick and then decreases very slowly. The graph is like this And let's assume the maximum occurs when $x=20$. I want to use taylor expansion to claim why in ...
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2answers
87 views

Why is the MacLaurin series proof for eulers formula $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $ valid?

The proof for this $$ e^{i\theta} = \cos(\theta) + i\sin(\theta) $$ using the MacLaurin series is all right for a high school level, but I dont understand why the series that has been derived for the ...
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39 views

Proof of Krull's intersection theorem with Taylor expansion

I took a commutative algebra course last semester (with Kaplansky's book), and I've learned about Krull's intersection theorem. In the course, we proved it without using Artin-Rees Lemma. I heard that ...
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1answer
37 views

Why does this work to shift a power series?

Problem: Find the Taylor series and the interval on which it is valid for $f(x) = \frac{1}{1-x}$ centered around $x=5$. The textbook's solution says to write $$\frac{1}{1-x} = \frac{1}{-4-(x-5)} = -\...
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2answers
130 views

Find Taylor series polynomial that gives uniform bound on error

The problem comes in two parts: Find an $\epsilon > 0$ such that for every $x\in[0,1]$ $$\left\lvert \sqrt{x}-\sqrt{x+\epsilon}\right\rvert \le \frac{1}{200}$$ We can show that $\left\lvert \...
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0answers
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How to determine the truncation error with products and quotients

If I have an equation given by $$\displaystyle Y = \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$ and I expand $a,b,c,d$ in a Taylor series, where $a$ is truncated at the $A^{th}$ order, $b$ is ...
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1answer
41 views

An alternative formula for a second order Taylor expansion?

I read in a book that the second order Taylor expansion of a function (around $x^0$) can be written as: $$f(x)=f(x^0)+\sum_{j=1}^n df(x^0)/dx_j*(x_j-x_j^0)+\sum_{j=1}^n\sum_{i=1}^nd^2f(x^1)/dx_idx_j*(...
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1answer
21 views

Tanh expansion problem

$$\frac{exp(-\beta e_k) - exp(\beta e_k)}{exp(-\beta e_k) + exp(\beta e_k)} = tanh(-\beta e_k)$$ In the context of mean field annealing I reached this equation but I am not sure how to expand tanh to ...
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Taylor Series coefficients

I have performed a Taylor Series expansion of a 2-D function in variables (y1,y2) and got something like: $f(y_1,y_2) = 3y_2 + 0.5y_1^2 + ...$ My question is that I would like this to "match" to ...
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1answer
26 views

Algebraic series, rational fraction of two variables in the form of polynomial

I come across the following claim: Let $y\in\mathbb{C}[[x]]$ be an algebraic series, that is, there exist $n\in\mathbb{N}^*$ $A_i(x)\in\mathbb{C}[x]$ for $i=0,...,n$ and $A_n(x)\neq 0$ such that \...
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3answers
75 views

Find the power series representation of $e^{-x^2}$

I know that the Maclaurin expansion of $e^x$ is $$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$$ But i'm not sure how to find the Maclaurin series here I tried this $$ f'_{(0)}=-2xe^{-x^2}=0 $$ And that ...
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2answers
54 views

Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives

I am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old series,...
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2answers
36 views

Computation of a complicated limit

Good morning to everyone! I don't know how to compute this type of limit... I got stuck at $arctan$. The limit is the following: $$ \lim _{x\to \infty }\left(\frac{\arctan \left(1-\cos \left(\frac{1}{...
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1answer
83 views

How to prove this integral equality?

The question is prove that$$ \lim_{n \to \infty} n^2 \left( \int_a^bf(x) \, \mathrm{d}x - \frac{b-a}{n} \sum_{i=1}^{n} f(a+(2i-1) \frac{b-a}{2n} )\right)= \frac{ (b-a)^2 }{24}\left( f'(b)-f'(a)\right)....
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3answers
45 views

Limit of the given expression.

For $x>0$,$$\lim_{x\to 0}((\sin(x))^{\frac{1}{x}}+(\frac{1}{x})^{\sin(x)})$$ is?. So now I calculated limits individually. Let $\lim_{x\to 0} ((\sin(x))^{1/x})=y$ thus I took log to get $\frac{1}{x}...
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1answer
24 views

Using Taylor Polynomial to Show How An Expression Of Only Real Numbers Can Be Approximated

I am studying for my graduate level GQE and looking at problems from old exams. The following question (from an unknown original source) reads: Suppose a,b,c and d are positive real numbers with a $&...
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2answers
51 views

Power series as approximation

I have to estimate the error when I approximate the function $$e^{\sin x}$$ to $$1+x+x^{2}+x^{3}$$ when $|x|<0.1$. I really don't know how to do because my teacher didn't teach me. But what I did ...
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2answers
42 views

Find the Maclaurin series of f(x)=(arctan(x)-x)/x^3

What I think I need to to do is find a general series expansion of the function and then derive term by term to get the Macaurin series...but I'm not quite sure how to expand this function. Any help ...
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0answers
26 views

Maclaurin polynomial of order 3? Order vs. Degree

I am doing some homework and came across a problem that asks: Find the Maclaurin polynomial of order 3 for f(x) = e^(-4x) When did some searching online, all searches came up as "...maclaurin ...
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1answer
112 views

new equation for $\int_0^ t e^{-x2} dx$? [closed]

fact! $$\int_0^ x e^{-x^2} dx$$ $$=e^{-x^2}\sum_{n=0}\frac{(2^n)x^{2n+1}}{{(2n+1)!!}}$$ Well the equation was new to me, when I derived by shear integration, and that is a cold HARD fact. The ...