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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4
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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
184 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'...
0
votes
2answers
32 views

zero of second order

I'm studying functions associated with a domain in the complex plane. In one paper that I'm reading, a particular function, $R(a, b)$, is discussed (with "$a$" varying and "$b$" fixed complex ...
1
vote
0answers
35 views

Any way to characterize this family of polynomials?

I have a family of polynomials generated by the recurrence relation $P_{n+1}(w) = (1+w)P_n ^{\ \prime}(w) -(3n-1 +nw)P_n(w) \\ P_1(w) =1$ The family is related to the Lambert $W$-function by its ...
1
vote
0answers
35 views

A geometric proof for the “small angle approximation” for the sine, cosine and tangent

How can be deduced the so called "small angle approximations" for sine, cosine and tangent, namely $\sin \theta \approx \theta$ $\tan\theta \approx \theta$ $\cos\theta \approx 1-\frac{\theta^2}{2}$ ...
0
votes
1answer
13 views

Taylor Series function seperation.

Say we have a function $$ F= \frac{g}{h} $$ And we want to expand it with Taylor series keeping only second grade terms. How do we know when to expand $F$ directly or $g$ and $ \frac{1}{h} $ ...
0
votes
0answers
11 views

Taylor series of complex function

$f(z) = \frac{2}{z^2-1}$ at $z = i$ My solution: $t = z - i$ $z = t + i$ $\frac{2}{z^2-1} = -2\frac{1}{1-(t+i)^2} = -2\sum_{n=0}^\infty (t+i)^{2n} = -2\sum_{n=0}^\infty z^{2n}$ Where is my ...
0
votes
3answers
50 views

Limit of $\lim_{x \searrow 0} \frac{\sqrt{1 + 2x + 5x^2} - e^{3x}\cos(2\sqrt x)} {\sin(x^2)} $ with Taylor expansion

How to calculate the limit of the following function using Taylor series: $$\lim_{x \searrow 0} \frac{\sqrt{1 + 2x + 5x^2} - e^{3x}\cos(2\sqrt x)} {\sin(x^2)} $$ I know how to get the series for $\...
1
vote
2answers
54 views

Calculating $\lim \limits_{x \to 0} \frac{\cos x\sin x-x}{\sin^3 x}$ using taylor series

I am given the following limit, and need to calculate it using taylor series: $$\lim \limits_{x \to 0} \frac{\cos{x}\sin{x}-x}{\sin^3{x}}$$ $$\sin{x}=x-\frac{x^3}{3!}+\frac{x^5}{5!}...$$ $$\cos{x}=...
2
votes
1answer
28 views

Using a taylor polynomial to approximate $cos(\frac14)$ with an error no more than $10^{-12}$

So the lagrange remainder is given by: $$R_n(x)=\frac{f^{n+1}(c)}{(n+1)!}(x-a)^{n+1}$$ We want $cos(\frac14)$ and we can do it around a=0. We know that $f^{n+1}$ is either $\pm$ cosx or $\pm$sinx, ...
0
votes
0answers
20 views

What exactly is the variable of Laplace transform

I try to find the solution for a hard differential equation. I could not solve it with any orthodox method. However, if I use Laplace transform and then replace its term with its Maclaurin series, it ...
1
vote
1answer
36 views

Find a Laurent expansion

I need to find Laurent expansion for function $$f(z) = \frac{iz^2 + 4iz + 4 +12 i}{(z^2+4)(z+2-i)}$$ for $2 < |z-i| < 3$. I start with division: $$\frac{iz^2 + 4iz + 4 +12 i}{(z^2+4)(z+2-i)} = \...
2
votes
3answers
76 views

Find the 8th derrivative of the function $h(x) = xe^x $using sequences

How do you find the 8th derivative of $h(x) = x e^x $ without doing it "manually". I know that $\displaystyle e^x = \sum_{i=0}^n \frac{x^n}{n!} $ so that $\displaystyle h(x) = x \sum_{i=0}^n \frac{x^...
1
vote
1answer
43 views

Calculus 2 - Prove Disprove - convergence of Taylor series

I got this question regarding properties of Taylor series. I'm stuck on the second question, I believe it is true since the area of convergence for X is affected by the coefficient and it is not ...
1
vote
1answer
15 views

About the set of $x$ values at which the Taylor series of $f(x)$ converges to $f(x)$

Let $f(x)$ be a function (for simplicity, let us assume that it is defined on $\mathbb{R}$ and infinitely differentiable), and $T$ the Taylor series of $f$ at $x=a$, with interval of convergence $I$. ...
1
vote
3answers
39 views

Asymptotic expansion of ratio function

I want to expand the following function: $$ f(x)=\frac{1}{(1-e^{-x})} $$ $f(x)$ can be rewritten as $$ f(x) \sim \frac{1}{x-x^2/2 + x^3/2/3} $$ But I want to express big-oh notation such that $$ f(...
4
votes
1answer
23 views

Relationship between taylor series and geometric series

To find the taylor series of a function you would usually use the formula $\sum_{n=0}^{\infty}\frac{f^{n}(c)}{n!}(z-c)^n$. However when computing the taylor series for $f(z)=\frac{1}{z+3}$ about $z=1$...
1
vote
1answer
34 views

Taylor expansions to approximate multivariable functions

The way this question is phrased is confusing me more than the question itself, so I will quote it how it is written in my book: "Using Taylor's theorem, find linear and quadratic approximations to ...
0
votes
2answers
46 views

Perturbation: compute an approximation to the solution of the equation $y+\epsilon\sin y=x^2$

Compute approximation to the solution of the equation $y+\epsilon \sin y=x^2$ using perturbation method. Assume that terms involving powers of $\epsilon$ of order 3 or more can be ignored. So far I ...
0
votes
3answers
56 views

Prove this for any $k>0$

Prove that $k!>(\frac{k}{e})^{k}$. It is known that $e^{k}>(1+k)$. So if we multiply $k!$ on both sides, we get $k!e^{k}>(k+1)!$. Also $k^k>k!$. Now how to proceed ?
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0answers
20 views

Landau notation related question

Hi :) Just a quick question here. When you put $cos(x)$ into wolframalpha, it says that the taylor series expansion about $x=0$ is $1 - \frac{x^2}{2} + \frac{x^4}{24} + O(x^6)$. My question is, how ...
1
vote
1answer
25 views

Proof of a lemma that is needed to derive Stirling's approximation

How do I prove that $$\left\lvert \log(1+x)-x+\frac{x^2}{2}-\frac{x^3}{3} \right\rvert \le \frac{x^4}{4}$$? (for $x \in (0,1) $) We used this result to derive the first order Stirling formula.
1
vote
1answer
50 views

Jacobian matrix and Taylor expansion

Let $\mathbf{W}(\alpha)$ be a matrix which depends to parameter $\alpha$ and let $\mathbf{f}$ be a vector. I want to approximate $\mathbf{W}(\alpha+\Delta \alpha)\mathbf{f}$ using Taylor expansion. ...
0
votes
1answer
29 views

Integral approximation of functions not defined for x=0

I have to approximate with an error less than 0.1 this integral: $$ \int_1^2 \exp\left(-\frac{1}{x^2}\right)\,dx $$ I understood I have to use the Taylor series, then prove that is uniforme ...
1
vote
1answer
30 views

What is the first order taylor expansion of a matrix to scalar function?

wikipedia says that if the real function f has the Taylor expansion: $f(x) = f(0) + f'(0) \cdot x + f''(0) \cdot \frac{x^2}{2!} + \dots$ then a matrix function can be defined by substituting x by a ...
-4
votes
2answers
114 views

Identifying $\sum\limits_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{k+6}$ [closed]

I'm trying to prove this equality. $$\sum_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{(k+6)} {=} e^x(x^5-5x^4+20x^3-60x^2+120x-120)+120$$ posted by: http://math.stackexchange.com/q/832368. How do I get ...
0
votes
0answers
41 views

Taylor Series in any vector space?

I am working through Alexander Kirillov, Jr.'s An Introduction to Lie Groups and Lie Algebras, and on page 29 he does something I find puzzling. He claims that, since the exponential map is a local ...
0
votes
1answer
52 views

Understanding an apparent contradiction from naively applying Taylor's theorem and Fubini's theorem together

Suppose $f(x)$ is a bounded, $\mathcal{C}^{\infty}$ function on $\mathbb{R}$ for which the integral $$I = \int_{0}^{\infty}f(x)\ dx,$$ exists. Taylor's theorem implies $f$ admits a MacLaurin expansion ...
1
vote
2answers
42 views

Formula for the general term of the Taylor series of $\tan(x)$ at $x = 0$

Today we were taught different expansions; one of them was the series expansion of $\tan(x)$, $$\tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15} + \cdots .$$ So, with curiosity, I asked my sir about next term. ...
2
votes
0answers
33 views

Power series expansion of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ and $z\mapsto \tan z$

Determine the power series expansion and radius of convergence of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ around $0$ with $t\in\mathbb C$. Determine the radius of convergence and the first three non-...
3
votes
0answers
40 views

Are these Infinite Series Representations of Special Functions?

I am not sure how to google the answer for this question. Anyway, in trying to compute the velocity of a charged particle in an electromagnetic field, I came across these two infinite series (...
0
votes
1answer
30 views

Find limit using Maclaurin power series

I encountered the following problem: $$ \lim_{x\to 0} \frac{x-\ln(1+x)}{x-\arctan x} $$ I expanded $ \arctan x $ in the denominator up to the fifth term and get the following: $$ x - \left(x - \...
0
votes
1answer
13 views

Get transfer function of a nonlinear diff. equation

I have this equation: $$\frac{\partial v}{\partial t} = -g + c\left(u(t) - v(t)\right)^2$$ g and c are constants. u(t) is my input and v(t) is my output. I need to reach the transfer function $\frac{...
2
votes
0answers
23 views

Maclaurin series remainder in Ordo-form

I encountered a problem where two of the terms are the following: $$ \cdots+\frac{1}{2!}(x-\frac{x^3}{3!} + \omicron(x^5))^2 + \frac{1}{3!}(x-\frac{x^3}{3!} + \omicron(x^5))^3 $$ It's suggested that ...
0
votes
1answer
28 views

Macularian series for natural log

So, I know that $$ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ... $$ Am I right in assuming that I can derive to follow by a subtitution of $-x$ $$ln(1-x) = -x - \frac{x^2}{2} - \...
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votes
0answers
20 views

The “overdamped” approximation: deleting the higher order term in an ODE

Say we have an ode of the form $$ \epsilon \ddot{x} + a\dot{x} + b x = 0 $$ If $\epsilon$ is small enough the approximation $$ a\dot{x} + b x = 0 $$ is often done in physics; in fact, I'm interested ...
0
votes
1answer
38 views

How can I prove that it is an Entire Function

Prove that if $$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$ ...
0
votes
1answer
30 views

Taylor series integration

I am having trouble with the following question: Integrate the Taylor series $$e^{(-t^2)} = \sum^\infty_{n=0} \frac{(-t^2)^n}{n!}$$ term-by-term to obtain the Taylor series for erf (error function) ...
1
vote
1answer
47 views

Approximate integral using Taylor Series

I have to approximate this integral with an error lesser than 0.1 using Taylor Series. This is the integral: $$\int_0^1 \arctan(\frac{1}{x^{10}}) dx$$ If I understood, I have to determinate the Taylor ...
0
votes
1answer
21 views

Limit of a function containing $\Psi(x)$

The taylor series expansion of the function $$f(x)=\ln(1+x)$$ around zero is: $$f(x)=\sum_{k=1}^\infty\dfrac{(-1)^{(k+1)}}{k}x^k$$ Putting $x=1$ we have the alternating series: $f(1)=1-\dfrac{1}{2}+\...
0
votes
3answers
34 views

How to differentiate the Taylor expansion?

We know the Taylor expansion of $f(x)$ at $a$ is and let it be $g(x)$, then $$g(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\ldots$$ My question is, is it ...
0
votes
2answers
23 views

First two terms of the Taylor series of the $n$-th iterated of a holomorpic function

Let $G$ be a region in $\mathbb{C}$ (i.e. $G ≠ \emptyset$ is simply connected and open), with $0 \in G$. Let $f: G \to G$ be a holomorphic function that's Taylor series (around $0$) has the shape $z + ...
1
vote
0answers
29 views

What is the general Taylor Expansion for the following function of a function.

guys. I am stuck with a general form of Taylor Expansion of following function, which is defined as a function of a function: $$F(X(t+h))-F(X(t))=[X(t+h)-X(t)]\frac{dF}{dX}(X(t))+\\\frac{1}{2}[X(t+...
0
votes
1answer
51 views

Closed form of a series

I am looking for a closed form of the following convergent series: $$\sum_{n=0}^\infty \frac{(-\lambda^2)^n}{(6n+i)!}$$ For the case of $i=0$, the answer is ready, but when $i=1,2,3,4,5$, everything ...