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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3
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67 views

Show that $f$ is identically zero in $\mathbb{C}$

Let $f$ be holomorphic in $\mathbb{C}$. Prove that if $|f(z)| \leq M|z|^{\alpha}$ with $0 <\alpha <1$, then $f$ is identically zero in$\mathbb{C}$ I know that $f$ has a Taylor expansion $f(...
0
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0answers
24 views

How to obtain this expansion?

My goal is to find an expansion in powers of 1/ρ (and its first 2 or 3 terms) of the quantity \begin{equation} F(\rho,\mu,\nu)=(2n+1)E_n(\rho)E_n(\mu)E_n(\nu)I_n(\rho),\quad \rho \ge h_2 \end{equation}...
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0answers
32 views

Different Alternate Representations of Functions

Could someone please point out a source with detailed steps / other pointers for different alternate representations of functions? For example, I know of two such ways 1) Taylor Series Expansion 2)...
3
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2answers
24 views

Derivation of Taylor expansion with the $a$ term

If we have $$f(x) = \sum_{n=0}^\infty a_n x^n$$ The $k$th derivative is $$f^{(k)}(x) = \sum_{n=0}^{\infty} a_{n+k} \frac{(n+k)!}{n!} x^n$$ Which also means that $$f^{(k)}(0) = k! a_k$$ Implying ...
0
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0answers
32 views

Evaluation of $\exp\left(a\frac{d^2}{dx^2}\right)f(x)$

I know that \begin{align*} \exp\left(a\frac{d}{dx}\right)f(x)=f(x+a)\,, \end{align*} by comparing the Taylor expansions of both sides ($f(x)$ is an arbitrary function). However, if I have, where $f(...
1
vote
2answers
52 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 ...
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0answers
20 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
vote
1answer
26 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 ...
0
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1answer
49 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
34 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
30 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
votes
2answers
44 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
votes
3answers
69 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(...
0
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0answers
19 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!}\...
0
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2answers
55 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
42 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
vote
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
votes
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
vote
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
25 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
votes
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
votes
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). ...
0
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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
votes
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
votes
3answers
71 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
votes
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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vote
3answers
32 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\...
0
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0answers
23 views

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

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
26 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: $$\...
0
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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
votes
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): ...
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
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: $\sqrt{3}/2$ ...
0
votes
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
votes
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
vote
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
votes
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
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
votes
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
votes
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=...