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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1
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1answer
18 views

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

I would like to prove the following: $$\sin\left(2\pi\sqrt{n^2+(-1)^{n}} \right)=\dfrac{(-1)^{n}\pi}{n}+\mathcal{O}\left( \dfrac{1}{n^2}\right). $$ My attempt: \begin{align*} \sin\left(2\pi\sqrt{n^...
5
votes
1answer
26 views

Show $\cos\left( \pi n^{2}\ln\left(\frac{n}{n-1} \right) \right)=(-1)^{n+1}\frac{\pi}{3n}+\mathcal{O}\left( \frac{1}{n^2}\right) $

I would like to show : $$\cos\left( \pi n^{2}\ln\left(\dfrac{n}{n-1} \right) \right)=(-1)^{n+1}\dfrac{\pi}{3n}+\mathcal{O}\left( \dfrac{1}{n^2}\right) $$ by starting from the left side and get the ...
1
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1answer
22 views

Show that $(-1)^{n}\left( (n+1)^{\frac{1}{n+1}}-n^{\frac{1}{n}}\right)=\mathcal{O}\left(\frac{\ln(n)}{n} \right) $

I would like to show: $$(-1)^{n}\left( (n+1)^{\dfrac{1}{n+1}}-n^{\dfrac{1}{n}}\right)=\mathcal{O}\left(\dfrac{\ln(n)}{n} \right) $$ Here is my attempt \begin{align*} (-1)^{n}\left( (n+1)^{\dfrac{1}{...
2
votes
3answers
46 views

Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
1
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0answers
22 views

Taylor expansion with several variables

It is known that for $f:\mathbb{R} \rightarrow \mathbb{R}$, $$f(a + h) = f(a) + f'(a)h + O(h^2)$$ Is there a similar expression for $f:\mathbb{R}^n \rightarrow \mathbb{R}$? i.e., something like $$f(...
2
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0answers
33 views

Taylor series of a square matrix

Let $A$ be a constant square matrix, $\Delta t$ is a scalar. I would imagine a taylor series of its exponential like this: $$e ^{A\,\Delta t} = I + \sum_{n=1}^\infty \frac{(A\,\Delta t)^n}{n!} $$ ...
2
votes
1answer
36 views

Bounding the absolute error of the linear approximation by $|E|\le\frac{n^2M}{2}\|\mathbf h\|^2$

Let $f: D\subseteq \Bbb R^n \to \Bbb R$ be a $C^2$ function. I'm trying to show that the absolute value of the error of the first order Taylor approximation of $f(\mathbf x+\mathbf h)$ is bounded ...
0
votes
1answer
20 views

Show that $\frac{(-1)^{n}}{\left(\ln(n)+(-1)^{n}\right)^{2}}=\frac{(-1)^{n}}{\ln^{2}(n)}+v_n\quad \left( v_n\sim -\frac{2}{\ln^{3}(n)} \right) $

I would like to show that : $$\dfrac{(-1)^{n}}{\left(\ln(n)+(-1)^{n}\right)^{2}}=\dfrac{(-1)^{n}}{\ln^{2}(n)}+v_n\quad \left( v_n\sim -\dfrac{2}{\ln^{3}(n)} \right)\\ $$ by starting from the left ...
0
votes
0answers
15 views

Proof regarding little o for vector field

I'm struggling with one part of a proof of a theorem. Let $\gamma(t) : A \subseteq \mathbb{R} \rightarrow \mathbb{R}^m$, with $\gamma \in \mathscr{C}^1(A)$ (hence $\gamma$ differentiable). ...
0
votes
1answer
20 views

Show $ \frac{(-1)^{n}}{n-\ln(n)}=\frac{(-1)^{n}}{n}+\mathcal{O}\left(\frac{\ln(n)}{n^{2}} \right) $

I would like to show that : $$ \dfrac{(-1)^{n}}{n-\ln(n)}=\dfrac{(-1)^{n}}{n}+\mathcal{O}\left(\dfrac{\ln(n)}{n^{2}} \right) $$ by starting from the left side and get the right side My proof: ...
5
votes
2answers
56 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)^{\...
1
vote
1answer
12 views

Show $(-1)^{n}\ln\left[ \frac{n(n+2)}{n^2-n+1} \right]=3\frac{(-1)^{n}}{n}+\mathcal{O}\left( \frac{1}{n^2}\right) $

I would like to show that : $$(-1)^{n}\ln\left[ \dfrac{n(n+2)}{n^2-n+1} \right]=3\dfrac{(-1)^{n}}{n}+\mathcal{O}\left( \dfrac{1}{n^2}\right) $$ by starting from the left side and get the right ...
2
votes
0answers
55 views

What is the proof that $\int e^{-x^2} \cdot dx$ is not elementry. [duplicate]

Is there a proof that gives the evidence there is no closed form for $\int e^{-x^2} \cdot dx$? or just because they were not able to find that elementry form for a long time of trying without any ...
-1
votes
1answer
20 views

What is the area between two first order taylor series approximations as they become closer to eachother

Let's say that $y=\sin{x}$. Then the first order taylor series approximation about $c$ is $g(x)=\sin{(c)}+\cos{(c)}(x-c)$. Note that this is also equivalent to the line tangent to the curve $\sin{x}$ ...
1
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3answers
29 views

Showing that the remainder term in Taylor's Theorem Converges to Zero

On pg. 110 of Rudin's Principles of Mathematical Analysis, it is shown that if $f$ is a real function on $[a, b]$ with $f^{(n)}(t)$ existing for every $t \in (a,b)$, then there exists some $x \in (a, ...
0
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0answers
15 views

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$ $o_i$ = observed $e_i$=expected (I removed $i$'s for ease) The solution is: $$G^2=2\sum o \log \frac{o}{e}$$ $$=2\...
0
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2answers
29 views

Solving a nonlinear equation $\sum_{z=0}^{s} \frac{(\lambda(l-x))^z}{z!} e^{-\lambda(l-x)}=p$

I would appreciate it if someone helps me with solving the following equation. Suppose $\lambda,l \in R^+$, $p\in[0,1]$, and $s\in N_{0}$. How can we find an $x\in [0,l]$, which satisfies the ...
0
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0answers
21 views

Upper and lower bound for maclaurin series of exponential function [on hold]

I have an algorithm like this: The algorithm and I want to find upper bound for O() notation and lower bound for Ω() notation. When I try debug the algorithm, It is maclaurin series but without 1,...
0
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1answer
25 views

Khinchin's Law of Large numbers proof unclarity.

This is the formulation: Let $X_n,n=1,2,...$ be independent, equally distributed random variables. $EX_k=a$(expectation) $k=1,2,...$. For this sequence of $X_n$ the law of large numbers applies: $$\...
3
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0answers
32 views

inverse function and maclaurin series coefficients.

i dunno if this is asked before, and i am not sure where to find this on the web or in textbooks. we are given a function (that is too hard to invert by solving for $x$): $$ y = f(x) $$ which has ...
2
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1answer
50 views

Multivariable Taylor Series

I would like to show the validity of the multivariable version of Taylor series expansion up to second-order terms (if possible without using one of the explicit forms for the remainder term): ...
2
votes
2answers
48 views

nature of the series $\sum (-1)^{n}n^{-\tan\left(\tfrac{\pi}{4}+\tfrac{1}{n} \right)}$

I would like to study the nature of the following serie: $$\sum_{n\geq 0}\ (-1)^{n}n^{-\tan\left(\dfrac{\pi}{4}+\dfrac{1}{n} \right)} $$ we can use simply this question : Show : $(-1)^{n}n^{-\tan\...
1
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0answers
28 views

Can I apply integration by parts to the integral $\int_{-\infty}^{\infty}\left[u'(x)|_{x=a_0}\right](x-a_0)v(x)dx$

Suppose, I have an integration $I=\int_{-\infty}^{\infty}u(x)v(x)dx$, where $u:X \to Y$ and $v: X\to Y'$ are $n^{th}$ order differentiable functions of $x$. Expanding $u$ around an arbitrary point $...
0
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0answers
16 views

Help with linearization using Taylor Series

If I sound rather clueless, it is because I am. I'm having trouble with linearizing the following non-linear system: $$ 2\frac {dy(t)} {dt} = -y(t) - 0.9u(t)³ + 1.4q(t) $$ Where u(t), q(t) are ...
2
votes
2answers
37 views

Find the first four nonzero terms of the Taylor series for $\sin x$ centered at $\frac{\pi}6$

Find the first four nonzero terms of the series for $f(x)$ centered at $a$, using the definition of Taylor series. $$f(x) = \sin(x),\quad a=\pi/6$$ I got this: 1st term: $1/2$ 2nd: $\sqrt{3}/2$ ...
1
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1answer
118 views

Taylor expansion of $f(x) = \frac{x}{x+3}\frac{1}{x-2}$ near $x=2$.

I am trying to Taylor expand the function $$f(x) = \frac{x}{x+3}\frac{1}{x-2}$$ around the point $x_0 = 2$. Clearly, the last factor explodes around this point, so I will try and expand that term. ...
0
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0answers
15 views

Taylors theorem for second variation

In Hilberts Methods of mathematical physics (p. 214), a functional $$J[\varphi] = \int_{x_0}^{x_1} F(x, \varphi, \varphi') \mathrm dx$$ is expanded by Taylor's theorem $$J[\varphi + \epsilon \eta] =...
1
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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
30 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
31 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
votes
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
vote
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
votes
1answer
57 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
votes
1answer
41 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
votes
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 ...
2
votes
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 ...
4
votes
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
vote
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
votes
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
votes
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
votes
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
votes
1answer
411 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
votes
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 ( * ) ...
1
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0answers
28 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 ...
-1
votes
1answer
65 views