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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1answer
29 views

Intuitive derivation of Taylor expansion?

I was looking up the derivations of Catalan numbers, and one derivation (probably the most famous) involves generating functions that leads to: $$C(x) = \frac{1}{2} (1 - \sqrt{1-4x})$$ And then this ...
0
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0answers
14 views

When can you “plug in” a function $g(x)$ directly into a taylor expansion of a function $f(x)$ to get the expansion of $f(g(x))$, specifics below

I have asked a couple of questions Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives Computing the taylor series of $f(x)=(x+2)^{1/2}$ around $x=2$ up to ...
1
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1answer
24 views

Is the mathematical syntax correct here (Taylor-polynomial)?

Say I'm supposed to create the $2^{nd}$ degree Taylor-polynomial of $f(x) = \cos x$ at $x_{0} = 0$ I'd like to know if the syntax is correct, how I solved this little task. We have defined the ...
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1answer
39 views

A problem related to Taylor series [on hold]

Prove that there exists a constant $C \in \mathbb{R}$ so that $$\sum_{k=1}^{N}\frac{1}{k}=\log{N}+C+O\left(\frac{1}{N}\right)$$
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0answers
31 views

Does Taylor series around point zero (maclaurin series) always exist for differentiable function?

For every differentiable function $f(x)$, is $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} \, (x)^{n}$ always true and can be written? The only thing we have to be concerned is just whether ...
0
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1answer
28 views

Difference between little o and big O notation in taylor expansion

I know I can say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+o(\Delta t) $$ But can I say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+O((\Delta t)^2) $$ In both cases, do I have to add that $\Delta t \...
-1
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2answers
42 views

want to check answer given in my book is correct or not [duplicate]

The problem is to find coefficient of $x^n$ using binomial theorem for rational index in the expansion of $$\frac{1}{1-x+x^2-x^3}.$$ In my book the answer is given as $$\frac14+\frac{n+1}{2}+\frac{(-...
3
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3answers
59 views

Taylor-polynomial of function $f(x) = e^{x}*\sin(2x)$

This is not homework, I'm asking to learn for an exam which I'll write in 2.5 months. Count the Taylor-polynomial 3th grade of the function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = e^{x}*\sin(...
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0answers
18 views

Is $\frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 < \frac{1}{2}\frac{da}{dt}(t) \Delta t^2$ always true for $\Delta t$ small than one?

We know that the taylor expansion of $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...
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2answers
53 views

Taylor-polynomial of $f(x)=\log(\cos(x))$

$f: (-\frac{\pi}{2}, \frac{\pi}{2}) \rightarrow \mathbb{R}, f(x) = \log(\cos x)$ Count the Taylor-polynomial $T_{2}(f, 0)(x)$ of the second degree of $f$ in $x_{0} = 0$ Alright because it was ...
1
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0answers
56 views

nature of series $\sum_{n\geq 0}(-1)^{n}u_n $

Let $(u_n)_{n\in\mathbb{N}}$ be sequence defined as follows: $$\left\{ \begin{cases} u_0\in\mathbb{R}^{+}\\\forall n\in\mathbb{N},\quad u_{n+1}=\dfrac{e^{-u_n}}{n+1}\\ \end{cases} \right\}$$ ...
0
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0answers
41 views

Rigorious formulation of approximation of integral as square for large 2nd derivative.

We know that the taylor expansion of $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...
1
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0answers
42 views

Taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ Is the taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$ at $t$ = $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = ...
5
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2answers
45 views

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

I would like to prove the following: $$\ln\left(\dfrac{n+(-1)^{n}\sqrt{n}+a}{n+(-1)^{n}\sqrt{n}+b} \right)=\dfrac{a-b}{n}+\mathcal{O}\left(\dfrac{1}{n^2} \right) $$ My attempt i tried this way ...
1
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1answer
19 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
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1answer
30 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
26 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}{...
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0answers
24 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
44 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: $$e ^{A\,\Delta t} = I + \sum_{n=1}^\infty \frac{(A\,\Delta t)^n}{n!} $$ ...
0
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1answer
22 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
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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). ...
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1answer
22 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: ...
2
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0answers
56 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
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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
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3answers
69 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 ...
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1answer
25 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 $\sin{x}$ ...
2
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1answer
38 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 ...
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3answers
30 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, ...
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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\...
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0answers
23 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: $$\...
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2answers
36 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 ...
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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2answers
51 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\...
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): ...
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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 $...
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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
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2answers
38 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: $\sqrt{3}/2$ ...
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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] =...
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
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
votes
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{...
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 ...