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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2answers
526 views

Lagrange remainder vs. Alternating Series Estimation Theorem: do they always give you the same error bound?

Given a function and its nth degree Taylor series approximation, we can use the Lagrange form of the remainder to get a maximum value of the error of approximation. If the series is also an ...
2
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0answers
33 views

Asymptotic expansion of elliptic integral

I am trying to find the first 2-3 terms of the asymptotic expansion in terms of 1/ρ of the elliptic integral \begin{equation} I_n(\rho)=\int_0^\frac{h_2}{\rho}\frac{t^{2n}/h_2^{2n}}{(E_n(t))^2\...
1
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0answers
31 views

Question About Cauchy Product;

Is there a formula to multiply many series (More than two) using the Cauchy product? If there isn't, please tell me how I can write this formula $ \left( \frac{1}{a-e^x} \right) ^{n+1}$as the ...
0
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1answer
32 views

Prove that this limit is the logarithmic derivative of the Riemann zeta function.

Prove the following limit: $$-\frac{\zeta '(s)}{\zeta (s)}=\lim_{c\to 1} \, \left(\frac{\zeta (c) \zeta (s)}{\zeta (c+s-1)}-\zeta (c)\right)$$ As a starting point I tried to enter this series ...
3
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0answers
38 views

Find the limit of a definite integral

A definite integral is defined as $$I(v,\theta)=\int_0^{\pi} e^{v[\cos(\theta-\phi)-1]}\sqrt{\dfrac{v \sin\phi}{\sin\theta}}d\phi$$ My question is how to show that $$\lim_{v\to \infty} I(v, \theta)=\...
0
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1answer
39 views

Find the general formula for this Taylor series

Again stuck on this stuff. I swear I had the right answer... 4 times in a row... and now I'm stuck with one attempt left and i'm afraid to try again I think what I am doing wrong, is I am missing ...
0
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0answers
48 views

Is there any approximation expression of finite sum of exponential taylor series

Is there any approximation expression of below? $$\sum_{i=0}^{n-1}\frac{1}{i!}x^i$$ $n$ is small like 5 or 10. What i finally like to do is finding $x$ which satisfy $\sum_{i=0}^{n-1}\frac{1}{i!}x^i=...
1
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2answers
27 views

Find the general formula for this Macluaurin series

I've tried looking at video examples from my e-book, khanacademy, I can't find anything to explain this. My homeworks tutorial problems are always really confusing, they use terms when the book uses ...
2
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1answer
59 views

Show : $(-1)^{n}n^{-\tan\left(\tfrac{\pi}{4}+\tfrac{1}{n} \right)}=\tfrac{(-1)^{n}}{n}+\mathcal{O}\left(\tfrac{\ln(n)}{n^{2}} \right)$

I would like to show that : $$(-1)^{n}n^{-\tan\left(\dfrac{\pi}{4}+\dfrac{1}{n} \right)}=\dfrac{(-1)^{n}}{n}+\mathcal{O}\left(\dfrac{\ln(n)}{n^{2}} \right)$$ My proof: Note that : \begin{...
3
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1answer
42 views

An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
0
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0answers
8 views

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 ...
0
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0answers
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 ...
0
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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 ...
4
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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 \...
1
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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'...
4
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0answers
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 ...
2
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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 ...
0
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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 ...
3
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1answer
414 views

TaylorSeries of complete elliptic integral of the first kind

I want compute $K(k)$ as a Taylor Series; $k\in\mathbb{R}$ and $\vert k \vert < 1$. Can someone help me? $$ K(k):= \int^{\frac{\pi}{2}}_0 \dfrac{dt}{\sqrt{1-k^2 sin^2t}} $$ Results so far: $$ K(k):=...
4
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3answers
68 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
1
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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 ...
0
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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)...
2
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2answers
142 views

Series expansion for integral including error function

$\DeclareMathOperator{\erfc}{erfc} \DeclareMathOperator{\Ei}{Ei} $ What is the series expansion of $f$ for small $q$? \begin{align} U(q) &= q e^{q^2}\erfc q\\ I(q,q') &= \int_0^{2\pi}...
0
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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 ...
2
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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)}+\...
0
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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 ...
3
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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 ...
1
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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 ...
2
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1answer
29 views

Local quadratic approximation

I wanted to implement some penalized regression parameter estimation algorithm by Fan&Li (http://sites.stat.psu.edu/~rli/research/penlike.pdf, section 3.3, [1]), but cannot catch the idea of some ...
2
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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 ...
1
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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. $f(x)=x^{1/3}$...
10
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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 ...
2
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5answers
1k views

What does it mean intuitively for a Taylor Series to be centered at a specific point?

I understand what a Taylor series is and how to find the Taylor series of a function. However I do not understand intuitively what it means to find a Taylor series for a specific function, centered at ...
1
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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 \...
1
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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}...
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 ...
1
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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 ...
0
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0answers
15 views

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-...
0
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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 ...
1
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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 ...
1
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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 ...
2
votes
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 ...
-6
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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 ...
3
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0answers
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 ...
0
votes
1answer
38 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)} = -\...
0
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0answers
13 views

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 ...
0
votes
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*(...