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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2
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1answer
38 views

Find Polynomial of order 10 for $f(x)=sin(x)$ near x=0

My work so far : I presume the answer should look more like a summation? Thanks!
2
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2answers
78 views

Finding the limit of: $\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$ using taylor polynomials

no solution provided so I was hoping someone would do a quick look over and make sure it looks ok. Finding the limit of: $$\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$$ ...
0
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0answers
13 views

Linearization of a function at a point

I have this delay differential equation $$\frac{dx}{dt}=a(x(t)-x(t-1))-b |x(t)|x(t)$$ and I have to make a linearization at the point $\left(\bar{x}(t),\bar{x}(t-1)\right)$, but I cannot figure out ...
1
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0answers
37 views

Singular expansion of an implicit function

In the book of Flajolet and Sedgewick (this context is not so important, though), the following argumentation is used: Let $y(z)$ be a function given implicitly by $y - \phi(z,y) = 0$, where $\phi$ ...
2
votes
2answers
47 views

How to solve asymptotic expansion: $\sqrt{1-2x+x^2+o(x^3)}$

Determinate the best asymptotic expansion for $x \to 0$ for: $$\sqrt{1-2x+x^2+o(x^3)}$$ How should I procede? In other exercise I never had the $o(x^3)$ in the equation but was the maximum order to ...
4
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3answers
78 views

Compute $\frac{d^ny}{dx^n}$ if $y = \frac{7}{1-x}$

I am wondering what is $\frac{d^n}{dx^n}$ if $y = \frac{7}{1-x}$ Basically, I understand that this asks for a formula to calculate any derivative of f(x) (correct me if I'm wrong). Is that related to ...
0
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0answers
16 views

Implicit Euler using Taylor

I was reading script about differencial equatations. More specific about schemes that help calculate them - implicit Euler. That method was analyzed using something similar to Taylor but i am not sure ...
0
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0answers
19 views

How to find similar convergence rates?

Consider the Taylor's series infinite summation of $\sin(x)$. Let $A_k=\sum\limits_{i=0}^k(-1)^i{x^{2i+1}\over (2i+1)!}$ (Series expansion of $\sin(x)$) I need a series $\{C\}_n$such that its ...
1
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2answers
64 views

Show that the Maclaurin series of $f(x)=1/\sqrt{1-x}$ holds for $x\in[0,1/2]$ using the Lagrange remainder theorem

Show that the Maclaurin series of $f(x)=1/\sqrt{1-x}$ holds for $x\in[0,1/2]$ using the Lagrange remainder theorem I can see that $$f'(x)=\frac12 (1-x)^{-\frac32}\text{ and }f''(x)=\frac12\frac32(1-...
3
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6answers
97 views

How to prove this Taylor expansion of $\frac{1}{(1+x)^2}=-1\times\displaystyle\sum_{n=1}^{\infty}(-1)^nnx^{n-1}$?

I came across this series of the Taylor Expansion- $$\frac{1}{(1+x)^2}=1 - 2x + 3x^2 -4x^3 + \dots.=-1\times\sum_{n=1}^{\infty}(-1)^nnx^{n-1}$$ But I have no idea how to prove this... Thanks ...
2
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1answer
20 views

Computing the taylor series of $f(x)=(x+2)^{1/2}$ around $x=2$ up to order 2 terms using binomial series

I can expand $(x+2)^{1/2}$ by taking $y=x+1$ and using the binomial series to find a power series representation of $f$: $$\sqrt{y+1}=\sum_{k=0}^{\infty}{1/2\choose k}y^k=1+\frac{y}{2}+\frac{\frac{1}{...
1
vote
1answer
30 views

How do I apply a Taylor expansion of this?

Given $$\frac{1}{r}\left(1+\frac{2\epsilon \cos\theta}{r}\right)^{-1/2}$$ I was told by using Taylor expansion I could get $$1-\frac{2\epsilon \cos\theta}{r}$$ with term of order $\epsilon^2$. Can ...
0
votes
1answer
75 views

How to derive a Taylor series from the ones we know ($\cos x$, $\sin x$, …)

If we know the Taylor expansion for the $\cos(x)$ function around $0$, how can we use it to derive the Taylor expansion of a similar function ($\cos(x+π/4)$) around $0$? I do know how to get the ...
0
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1answer
70 views

developing to maclaurin Series $f(x)=\frac{2x+3}{x^2 -4x+5}$ on $x=2$

$$f(x)=\frac{2x+3}{x^2 -4x+5}$$ on $x=2$. My solution: $t=x-2 $ => $x=t+2$ , we get: $f(t)=\frac{2t+7}{t^2+1}$ on $t=0$. then: $(2t+7)\sum_{n=0}^{\infty } {(-t^2)^n} = (2t+7)\sum_{n=0}^{\infty }{(...
0
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2answers
47 views

Show complex equation of closed curve integral

I need to show this equation: $$\frac{1}{2ia} \cdot \oint _{\gamma } \frac{e^{iz}}{z-ia}dz = \frac{e^{-a}}{2ia} \cdot \oint _{\gamma } \frac{1}{z-ia}dz$$ I have an hint to using Taylor. I have no ...
7
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4answers
106 views

Proof of Leibniz $\pi$ formula

I found the following proof online for Leibniz's formula for $\pi$: $$\frac{1}{1-y}=1+y+y^2+y^3+\ldots$$ Substitute $y=-x^2$: $$\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots$$ Integrate both sides: $$\...
0
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1answer
28 views

Is it true that $(\Phi_{-t})_*w\circ \Phi_t=\sum_{n=0}\frac{t^n}{n!}\mathcal L^n_vw$?

Let $v,w$ be vector fields and let $\Phi_t$ be the flow generated by $v$, that is $\frac{d}{dt}\big|_0\Phi_t(x)=v(x)$. The Lie derivative of $w$ in direction of $v$ is usually defined as $\mathcal ...
1
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1answer
39 views

$\Pi_{n=0}^\infty (1-a_n)>0$ if and only if $\sum a_n < \infty$.

Let $a_n$ be sequence in (0,1). $\Pi_{n=0}^\infty (1-a_n)>0$ if and only if $\sum_{n=0}^\infty a_n < \infty$. First I considered $\sum log(1-a_n)$ and tried to find sum inequality. I ...
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2answers
49 views

Finding certain coefficients in Taylor expansion of $ \log(1 +qx^2 + rx^3)$

This exercise is part of the STEP $3$ paper from $2014$. At a certain point in the problem, we 're supposed find $a_n$ for $n = {2,5,7,9}$ where $a_n$ is the coefficient of $x^n$ in the series ...
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0answers
30 views

Integration in an inequality

Does integrating on both the sides of inequality with the same upper and lower limits with respect to same variable somehow affect the inequality. I saw an example lets say, Sin x < x ,x>0 ...
0
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2answers
33 views

How do I find the radius of convergence for $\sum_{n=0}^{\infty}\frac{1}{\sqrt{n}}z^n$?

I'm a little unsure about methods on finding the radius of convergence of a function. It would be great to get some help on how to approach these kinds of problems.
1
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1answer
40 views

How to find Taylor series when $x_0=0$ and radius of convergence for $\frac{x}{1+x}$ for $f:(-1,\infty)$

Through the taylor series formula: $$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\dots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$ I've got that $f(x)=x-x^2+x^3-x^3\dots$ however my teacher claimed ...
-1
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3answers
67 views

Is the following is true? If that so, give me a proof. $-log(1-x)=log(1+e^x)$??

Is the following is true? If that so, give me a proof. $$-log(1-x)=log(1+e^x)?$$ Give me some value where this equality holds. I dont think so it will be same. Because, $$(1-x)^{-1}=1+x+x^2+x^3+\...
1
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1answer
53 views

Calculus of rank three tensor

Let $A(\alpha)$ be a matrix that depends to vector parameter $\alpha$. I want to approximate $A(\alpha+\Delta\alpha)$ using Taylor expansion. My work: $$ A(\alpha+\Delta\alpha) \approx A(\alpha)+\...
1
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1answer
43 views

How to find taylor polynomial of a function with two variables?

Find the second order Taylor expansion about the point (1,-2) of the function $f(x,y) = (x^2 + y)e^{xy}$. I begin by computing the matrix of partial derivatives of f. $Df(x,y)=(2xe^{xy}+e^{xy}y(x^2+...
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0answers
13 views

How do I specify the input to a Volterra series including the kernels?

I have a series of dependent and independent variables. I would like to model their relation using a Volterra/Wiener series. How do I specify: $h_n$ $a$, $b$ Input vector $x_n$ The kernels for each ...
1
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1answer
31 views

Does there exist a kernel concept for Taylor expansions?

In the sense of kernel which one can use when weighing coefficients together in Discrete Fourier Transform (Dirichlet and Fejér kernels maybe most well known). Here are some examples of such kernels. ...
2
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0answers
25 views

How to find the asymptotic expansion of $\int_{-\infty}^{y} e^{-x^2/2}/\sqrt{2\pi} dx$ where $x \in N(0,1)$?

I realize the function inside the integral is the pdf of a normally distributed random variable x, but am unsure how to use this to solve the problem. I am trying to relate it to the inverse of the ...
2
votes
1answer
42 views

Multidimensional taylor series $sin (x^3y^2) $

A homework of mine was to compute the Taylor series of $f(x,y)=\sin(x^3y^2)$ around $(0,0)$ to the 25th order. I assumed, as $\sin(z)=\sum\limits^{\infty}_{k=0}(-1)^k\frac{z^{2k+1}}{(2k+1)!}$, that I ...
2
votes
2answers
43 views

Why my calculations aren't right? (Maclaurin series)

Good evening to everyone! I tried to calculate $ \cos\left( x- \frac{x^3}{3} + o(x^4)\right) $ using the MacLaurin series but instead of getting the final result equal to $1 - \frac{x^2}{2}+\frac{3x^4}...
2
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0answers
37 views

How to define hypergeometric function ${}_1 F_1(-n+1;-n+1;z)$ for $n$ positive integer

Consider a truncated Taylor series of the exponential function to approximate $e$: $$ E(n) = \sum_{k=0}^{n-1} \frac{1}{n!} $$ I thought of computing this using the hypergeometric finite series $_1 F ...
1
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1answer
49 views

How to do a Taylor expansion of a vector-valued function

Let $f:\Bbb R^2\to \Bbb R^2$ be given by $$f(x,y):= \left(e^x\sin(x+y),e^{y-x}\tanh(y)\right)$$ Find the second-order Taylor expansion of $f$ about (x,y)=(0,0)$. I know how to find the Taylor ...
0
votes
1answer
29 views

taylor series without dissipation

I need help. Determine the Taylor series about the point $(0,-1, 1)$ of $f(x, y, z) = z^3 - 3z^2 + x^2 + 4yx + 2y + 2z + 16$. Note. You must not derive. Thx
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0answers
22 views

Taylor series complex variable [closed]

Hello people i need two series of: $2z^2 - 3z$ with center at $z=-i$-----------the final answer must be something like this: $\sum(z+i)^n$ and the other one is: $(z^2 - 1)/(z)$ with center at $ z=...
1
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1answer
42 views

Taylor series for $g(z)=\frac{(z^2 - 1)}{(z)}$ [closed]

i need to find the Taylor series for $g(z)=\frac{z^2 - 1}{z}$ centered at $z=5$, can somebody help me please? Thanks!! EDITED: The answer must be something like this: Summatory of An(z - 5)^n from ...
1
vote
1answer
52 views

When is $1-(1-p)^n \sim pn$

Let $0<p=p(n)<1$ with $p=o(1)$. For which $p$ is it true that $1-(1-p)^n \sim pn$? With $\sim$ I mean that they are asymptotically the same, so $\frac{1-(1-p)^n}{pn}\rightarrow 1$, or at least ...
1
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0answers
73 views

Maclaurin Expansion of $\ln(3+x)$

I'm currently evaluating a simple Maclaurin expansion, the confusion I have with is why the expansion of this function is constructed to be: $\ln\left[3\left(1+\dfrac{x}{3}\right)\right]$ as opposed ...
1
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0answers
27 views

Taylor expansions for functions of several variables

I need help with this question. a) Determine the Taylor expansions at the origin up to the square Terms of $f(x, y, z) = \cosh(x) - \sin(yz) - xy(z - 1)^7$ and $g(x, y) = e^{-y}/(1-x^2)$. b) ...
1
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0answers
43 views

“Binomial Expansion” with a Complex Exponent

I want to expand, $$ (X + Y)^s $$ for $ X,Y \geq 0 $ and $ s \in \mathbb{C} $. Into something of the form, $$ \sum_i X^{p_i} Y^{q_i} $$ for $ p_i, q_i \in \mathbb{C} $. I am doing this to expand the ...
2
votes
1answer
122 views

Prove $\sqrt{1+x}$ can be represented by a power series

I need to show that $\sqrt{1+x}$ can be represented as a power series. I need to prove the equality between the function and its Taylor series, not to prove that the Taylor series of the function, $\...
2
votes
1answer
47 views

Singularities in the complex plane and expansion of Taylor/Laurent Series

The function f(z) = $\frac{\cosh(z-3i) -1}{(z-3i)^{5}}$ has one singular point in $\mathbb{C}$. I understand that the singular point is an isolated singularity at 3i, and I know there are certain ...
6
votes
2answers
68 views

Taylor Series as a linear operator $T:C^{k} (\mathbb{R} , \mathbb{R}) \to C^{\infty} (\mathbb{R} , \mathbb{R})$?

Can the Taylor series be thought of as either a linear operator $T: C^{k} (\mathbb{R} , \mathbb{R}) \to C^{\infty} (\mathbb{R} , \mathbb{R})$ given by $$ Tf=\sum^{k}_{n=0} \frac{f^{(n)}(a)}{n!}(x-a)^{...
4
votes
2answers
75 views

Calculating a limit with series

Good evening to everyone. I have a limit that gave me a lot of trouble and I couldn't find a way to solve it. I tried solving it with series but I couldn't arrive at a result. $$ \lim _{x\to 0+}\left(\...
7
votes
3answers
183 views

Elementary Proof of Ramanujan Master Theorem

I was searching for an elementary proof of the Ramanujan Master Theorem and I found a page from Ramanujan's Notebook on wikipedia which contained the proof. I think that it has some gaps, so can ...
2
votes
0answers
73 views

Taylor series around $x = 0$ of $f(x) = \int_0^x\frac{dy}{1+y^4}$

Find the Taylor series around $x = 0$ of $f(x) = \int_0^x\frac{dy}{1+y^4}$ and its radius of convergence. It seems like one ought to take the Taylor series of the integrand, and then integrate the ...
1
vote
0answers
31 views

Consistency in the definition of cross cumulants

Suppose that I have an $n\times 1$ random vector $X=(X_1,X_2,\ldots,X_n)'$. For $\xi=(\xi_1,\ldots,\xi_n)'\in\mathbb{R}^n$, we can define the familiar generating functions $$ M_X(\xi)=E\Big[\exp\Big(\...
2
votes
2answers
51 views

Intuition for polynomial bases

In my linear algebra course I stumbled upon the following observations. We have some function $f: \Bbb{R} \to \Bbb{R}$, $f = f(x)$. $f(x)$ may be composed of elementary functions or not, but in ...
5
votes
3answers
161 views

Find the Taylor series of $f^{\circ n}=\underset{n\text{ times}}{\underbrace{f\circ f \circ f\ldots \circ f}}$

Define $f(x)=ln(1+x)$. Then $f^{\circ 2}(x)=ln(1+ln(1+x))$, and $f^{\circ 3}(x)=ln(1+ln(1+ln(1+x)))$, etc. Find the Taylor series of $f^{\circ n}=\underset{n\text{ times}}{\underbrace{f\circ f \circ ...
0
votes
2answers
19 views

Taylor Approximations to avoid loss of significance

I am supposed to use taylor approximations to avoid loss of significance for the following functions: a) $f(x)=\frac{e^x-e^{-x}}{2x}$ b) $f(x)=\frac{log(1-x)+x*e^{x/2}}{x^3}$ and then find $\lim_{...
1
vote
1answer
47 views

Geometric proof of expansions of series

I have read that Barrow had proved the fundamental theorem of calculus. I have read that proof and its a good. Further I know Newton had derived the sine and cosine series. His methods obviously didn'...