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.

learn more… | top users | synonyms (2)

-4
votes
1answer
58 views

new equation for $\int_0^ t e^{-x2} dx$? [on hold]

fact! $$\int_0^ x e^{-x^2} dx$$ $$=e^{-x^2}\sum_{n=0}\frac{(2^n)x^{2n+1}}{{(2n+1)!!}}$$
1
vote
0answers
19 views

Maclaurin polynomial of order 3? Order vs. Degree

I am doing some homework and came across a problem that asks: Find the Maclaurin polynomial of order 3 for f(x) = e^(-4x) When did some searching online, all searches came up as "...maclaurin ...
1
vote
0answers
14 views

Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
0
votes
0answers
20 views

(Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
1
vote
1answer
40 views

Finding the Taylor series of arcsin(1-x)

I'm trying to calculate the Taylor series of $arcsin(1-x)$ about $x=0$. I'm having trouble because I can't compute the derivative there. I can see the correct solution on WolframAlpha (http://www....
2
votes
1answer
54 views

Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: \begin{equation} f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
-1
votes
0answers
13 views

Multivariable taylor series approximation

The function is of the form $$ F(X) = sum_{i=0}^n x_i*(c_i + ln(x_i/xt)) $$ where $ X = (x_1,x_2,x_3,...,x_n) $ $ xt = sum_{i=0}^n x_i $ $ c_i $ is a constant term for ith species I want to find ...
0
votes
2answers
32 views

Confusion about the different ways of writing Taylor Polynomials

For the sake of using a simple example, let's say I want to approximate $y=x^3$ with a second degree polynomial, and let's say I want to construct my polynomial around the point $x=4$. One way I ...
2
votes
2answers
30 views

Problem on series expansion and Bessel functions

One way to define Bessel functions is $$ e^{\frac{x}{2}(t-\frac{1}{t})}=\sum_{n=-\infty}^{+\infty}J_{n}(x)t^n. $$ How do I prove that? I can't see a way of writing the L.H.S. as a geometrical (or ...
2
votes
2answers
71 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: $$\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$$ ...
2
votes
1answer
32 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!
1
vote
2answers
512 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 ...
4
votes
3answers
74 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
votes
0answers
12 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 ...
0
votes
0answers
34 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
46 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 ...
0
votes
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
votes
0answers
14 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 ...
3
votes
6answers
94 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 ...
0
votes
1answer
30 views

Prove using taylor series $\lim\limits_{n\to \infty}\max\limits_{0<x\leq 1}\frac{d^n }{dx^n}\exp\left(\frac{-1}{x^2}\right) = \infty$ [closed]

Prove this using taylor series : $$\lim_{n\to \infty}\max_{0<x\leq 1}\frac{d^n }{dx^n}\exp\left(\frac{-1}{x^2}\right) = \infty$$ That is we need to provethat the derivative of $\exp\left(\...
1
vote
2answers
63 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-...
2
votes
1answer
19 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
27 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
73 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 ...
1
vote
2answers
36 views

Taylor Series in Fractional Calculus

I recently studied fractional calculus, namely the possibility to define fractional derivatives of some functions, like $$\frac{\text{d}^{1/2}}{\text{d}x^{1/2}}\ f(x) ~~~~~~~~~~~~~ \frac{\text{d}^{2/...
0
votes
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 }{(...
1
vote
2answers
42 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
votes
4answers
103 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
votes
1answer
27 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 ...
13
votes
1answer
313 views

Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the ...
1
vote
1answer
37 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 ...
1
vote
2answers
48 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 ...
1
vote
1answer
51 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
vote
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
votes
2answers
32 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
vote
1answer
39 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
vote
1answer
28 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. ...
-1
votes
3answers
66 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+\...
7
votes
3answers
170 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 ...
1
vote
1answer
37 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+...
1
vote
0answers
12 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
vote
1answer
48 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 ...
2
votes
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 ...
1
vote
1answer
50 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 ...
2
votes
1answer
40 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 ...
5
votes
3answers
158 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 ...
2
votes
0answers
36 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
vote
2answers
42 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}...
0
votes
1answer
26 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