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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$f(x)-T_{f,x_0,n}=o(x-x_0)^{n} \implies f(x) \sim T_{f,x_0,n}$?

I'm not sure about this. Consider a function $f$ and it's $n$ degree Taylor Polynomial in $x_0$ $T_{f,x_0,n}$. Considered the remainder function $R(x)=f(x)-T_{f,x_0,n}=o(x-x_0)^{n}$ Can I ...
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7answers
90 views

How to find the Maclaurin series for $(\cos x)^6$ using the Maclaurin series for $\cos x$?

Find the Maclaurin series for $(\cos x)^6$ using the Maclaurin series for $\cos x$ for the terms up till $x^4$. Here is what I've worked out: Let $f(x) = \cos x,\ g(x) = (\cos x)^6$. $$g(x) = (f(x)...
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2answers
38 views

How do I calculate the error bound for a Maclaurin series?

How many terms of the Maclaurin series of $f (x) = \ln(1 + x)$ are needed to compute $\ln(1.2)$ with an error of at most $0.0001$?
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4answers
54 views

Taylor series about 0 for $\frac{x^{2}}{e^{2x}}$

I got the following question: The Taylor series about 0 for $e^{x}$ is: $ e^{x} = 1 +x + \dfrac{1}{2!}x^{2} + \dfrac{1}{3!}x^3 + \dfrac{1}{4!} x^4 + \dots \qquad \text{for $x \in \mathbb{R}$} $ And ...
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4answers
50 views

Let $f(x)=\sum_{n=0}^{\infty}a_nx^n$ be convergent on $(-R,R),$ with $R>0.$ Prove that, if $f(x)=0$, $a_n=0$.

I'm stuck on a solution that our teacher gave to us. This is the exercise: Let $f(x)=\sum_{n=0}^{\infty}a_nx^n$ be convergent on $(-R,R),$ with $R>0.$ Suppose that $f(x)=0$ for all $x\in(-...
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2answers
52 views

Why is $(1+\frac{3}{n})^{-1}=(1-\frac{3}{n}+\frac{9}{n^2}+o(\frac{1}{n^2}))$ and how to get around the Taylor expansion?

Let be $(u_n)$ a real sequence such that $u_0>0$ and that $\forall n \in \mathbb{R}$: $$\frac{u_{n+1}}{u_n}=\frac{n+1}{n+3}$$ Let be $(v_n)$ a real sequence such that $\forall n \in \...
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1answer
28 views

Taylor Series from General Finite Difference Scheme

"For a 3-point stencil $[x_{i-1},x_{i+1}]$, we can write a generic expression as $\frac{\partial u}{\partial x}|_{x_i}=au_{i-1}+bu_i+cu_{i+1} + O(h^m)\qquad (1)$ where a,b, and c are unknowns to be ...
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1answer
265 views

Prove the identity $\cosh(2x)=\cosh^2(x)+\sinh^2(x)$ using the Cauchy product. [closed]

Prove the identity $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ using the Cauchy product and the Taylor series expansions of $\cosh(x)$ and $\sinh(x)$. The relations involving the exponential function are ...
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4answers
56 views

Remainder term for Maclaurin's $\sin x$ expansion

We know that for the Maclaurin's series $$\sum_{k=0}^{n}\frac{ f^{k}(0) }{(n+1)!}x^{k}$$ the remainder term is given by the following formula: $$R_{n} = \frac{\left | f^{(n+1)}(z) \right |x^{n+1} }{(n+...
4
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2answers
166 views

Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
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0answers
11 views

Inequality for a multivarialbe function?

For fixed $y\in \mathbb{C}^m$ and let $f$ be a fuction defined on $\mathbb{C}^m\times \mathbb{C}^m$ such taht $f(0,y)=1$ and $$\frac{\partial^n} {\partial x^n}f(x,y)=(i)^my^n{}f(x,y)$$ which means ...
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2answers
39 views

Evaluate the function $f(x)=\frac{x^2+3e^x}{e^{2x}}$ using Maclaurin series

$$x^2+3e^x=x^2+3\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}+O(x^n)=x^2+3\left(1+x+\frac{x^2}{2}+O\left(x^2\right)\right)$$ $$e^{2x}=\sum\limits_{n=0}^{\infty}\frac{(2x)^n}{n!}+O\left(x^{n+1}\right)=1+2x+...
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2answers
76 views

Show this assertion

I am stuck on the following task: Fix a vector $y \in C^n$ and let $f(x,y)$ be a function defined on $C^n \times C^n$ such that $f(0,y)=1$ and its $n$th partial derivative on $x$ satisfies $\frac{\...
3
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2answers
106 views

Ordinary generating function of powers of 2

Is there a good closed form expression for the generating function of the formal power series $$ A(z) := \sum_{n=0}^\infty z^{2^n} = z + z^2 + z^4 + z^8 + z^{16} + \cdots. $$ Is there a tractable way ...
0
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1answer
70 views

Solving $(1-x^2)y''-2xy'+a(a+1)y=0$

I need to find an even solution and an odd solution to the ODE $(1-x^2)y''-2xy'+a(a+1)y=0$ using a power series around $x=0$. I suspect I should use Frobenius, but not sure how to bring it to the ...
3
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3answers
67 views

Maclaurin Series Representation for $f(z)=\frac{z}{z^4+9}$

I need help finding the Maclaurin series representation for $$f(z)=\frac{z}{z^4+9}$$ I first tried to factorize $z^4+9$, but am I missing something? I could not figure out how to factorize this. Is ...
4
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2answers
48 views

How to compute $\lim_{x \to 0} (\frac{x^5 e^{-1/x^2}+x/2 - \sin(x/2))}{x^3})$?

I have a problem with this limit. I have no idea where is the problem. Can you correct my mistake? Thanks $$\lim\limits_{x \to 0} \left(\frac{x^5 e^\frac{-1}{x^2}+\frac{x}{2} - \sin(\frac{x}{2})}{x^3}...
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0answers
23 views

Connection between properties of taylor series and the function

Assuming I have a function $f(x)$ which at least for some $-R<x<R$ can be expanded in taylor series $$ f(x) = \sum_{n=0}^{\infty}c_n \frac{x^n}{n!} $$ are there any known connections ...
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2answers
31 views

Properties of the remainder function for Taylor polynomials

Considered $f$ differentiable at least $n$ times in $x_0$ and $P_{n,x_0}(x)$ the $n$ degree Taylor polynomial in $x_0$. Defined the Remainder function $R(x)= f(x)-P_{n,x_0}(x)$ I can't understand ...
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0answers
28 views

Edgeworth expansion of the sum of inid random variables?

This question relates to the asymptotic expansion for the distribution of sum of random variables using moments. Edgeworth expansion can be applied when the variables are independent and identically ...
9
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3answers
82 views

Prove that $\cosh^{-1}(1+x)=\sqrt{2x}(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+…)$

How can we prove the series expansion of $$\cosh^{-1}(1+x)=\sqrt{2x}\left(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+...\right)$$ I know the formula for $\cosh^{-1}(x)=\ln(x+\sqrt{x^2-1})$...
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1answer
46 views

How to show that $\frac{159999}{80000} +\frac{1}{100e^2} <\ln(e^2+\frac{1}{100} ) < 2+ \frac{1}{100e^2}$

I'm trying to show that $\frac{159999}{80000} +\frac{1}{100e^2} < \ln(e^2+\frac{1}{100} ) < 2+ \frac{1}{100e^2}$. I know I should do something with the first order taylor polynomial of $\ln(x)$ ...
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1answer
27 views

Finite expansion of this function

I had this result as finite expansion of this function $$\frac{1}{(1-x)^2}$$ to order n in neighborhood of 0: $$1+\sum_{i=1}^{n}{(x^i.(i+1)) }+0(x^n)$$ (where x tends to 0)is it true? And if yes ...
2
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1answer
64 views

How come $\frac{1}{\cos x} = 1 + \frac{x^{2}}{2} + o(x^{2})$ as $x \to 0$?

Since $$\cos x = 1 - \frac{x^{2}}{2} + o(x^{3})$$ as $x \to 0,$ we have $$\frac{1}{\cos x} = \frac{1}{1-\frac{x^{2}}{2} + o(x^{3})} = 1 + \frac{x^{2}}{2} + o(x^{3}) + o(\frac{-x^{2}}{2} + o(x^{3})).$...
2
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0answers
27 views

Relation between representation of a number in an integer base and Fourier series representation of a periodic signal

I am not a Mathematician - am just a software developer though I did some "Math" back in the day as part of my undergrad studies millions of years ago. Recently I had to revisit Fourier analysis of ...
3
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0answers
244 views

$4$- vector Taylor expansion, sign confusion

I've been presented with a function expansion which I'm told is correct but I can't figure out where the sign in the second term might be coming from. $$ e^{i\alpha(x_\mu + \epsilon \, n_\mu)} = ...
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0answers
37 views

External and internal multipole expansion for axisymmetric potential - the region of convergence.

Say, we have a system of electrodes exhibiting symmetry around a certain axis. We know the explicit expression for the potential on the axis $\phi (z)$. We want to find the potential for any point in ...
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1answer
77 views

Prove (or disprove) that $ \sum_{n=1}^\infty \frac{4(-1)^n}{1-4n^2} x^n = \frac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} - 2 $ for $ 0<x\leq1$

Just like title said, for $ 0 <x\leq1 $, prove/disprove: $$ \displaystyle \sum_{n=1}^\infty \dfrac{4(-1)^n}{1-4n^2} \cdot x^n \stackrel{?}{=} \dfrac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} - 2 $$ I ...
5
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0answers
58 views

Why does the taylor expansion of a nonlinear system of differential equations exist if it has continuous second order partial derivatives?

My textbook states that for a nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y)$$ The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever ...
0
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1answer
41 views

Exponential form for matrices

I'm trying to prove that for two commutative matrices $N$ and $M$, that $e^{N+M}=e^Ne^M$. I wrote using the binomial expansion and commutativity: $$e^{M+N}=\sum_{k=0}^{\infty}\frac{1}{k!}(M+N)^k=\...
2
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0answers
62 views

What is the lagrange remainder for $\sin x$?

What is the lagrange remainder for $\sin x$? $R_n=\frac{f^{n+1}(c)}{(n+1)!}x^{n+1}$ and $$\sin x = \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$$ So, is it $$R_{2n+1}=\frac{f^{2n+2}(c)}{(2n+2)!...
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2answers
33 views

Taylor series for multivariable functions

To expend the function of multiple variables $$ f({\bf x})=f(x_1,x_2,\dots,x_n):\mathbb R^n\to\mathbb R $$ in Taylor series around $\bf 0$, we have $$ f({\bf x})=f({\bf 0})+Df({\bf 0})\cdot{\bf x}+\...
1
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1answer
25 views

Proofs using the Taylor expansion, zero series and limits

If we have the function $f: (0,\infty) \rightarrow\ R$, $f(x) = e^{\frac{-1}{x^2}} $ Show that for $n≥0$, there exists a polynomial function $p_{(n)}(t)$ such that $f^{(n)} (x) =p_{(n)}(\frac{1}{x})f(...
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1answer
45 views

How to expand the Taylor series of functions of several vectors?

We know that the Taylor series expansion of the function of several scalars around zero is $$ f(x,y)=f(0,0)+f_x(0,0)\cdot x+f_y(0,0)\cdot y+\frac{1}{2!}f_{xx}(0,0)\cdot x^2+\dots $$ Then, how about ...
4
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4answers
249 views

Sine of argument with large n approximation

I have worked an integral and reduced the integral to $$\frac{n \pi+\sin\left ( \frac{n \pi}{2} \right )-\sin\left ( \frac{3 \pi n}{2} \right )}{2n \pi}$$ I want to show that for $$n\rightarrow \...
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0answers
65 views

Taylor Expansion of Power of Cumulative Log Normal Distribution Function - Show Lagrange Remainder tends to Zero

QUESTION I am looking to find a simplification of the expression below. I have attempted this using the Taylor series. The question then remains if we can show the Lagrange remainder goes to zero. I ...
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1answer
60 views

Why is this proof for Taylor's Remainder theorem not correct?

I am not exactly sure on how to post math equations in the question box so I have all my following information on a google document: https://docs.google.com/document/d/1vf20ZyLGQL-...
6
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1answer
8k views

Prove Taylor expansion with mean value theorem

On http://vapour-trail.blogspot.com/2006/03/brief-explanation-of-taylor-series-via.html one can find an hint at how to derive Taylor expansions from the mean value theorem. The process goes as follow....
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1answer
61 views

Taylor Series Polynomial Proof using Induction

If $f : \mathbb R \to\mathbb R $ is a polynomial function of degree $n$ with $a \in\mathbb R$. Show that the $n$-th Taylor polynomial $P_{f,a,n}$ of $f$ at $a$ is equal to $f$. I know that I need to ...
2
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1answer
35 views

Finding $w_1,w_2,b$ such that $w_2 \sigma(w_1 x + b) \approx x$

Let $\sigma(z) = 1/(1+e^{-z})$. How can I find $w_1, w_2, b$ such that $w_2 \sigma(w_1 x + b) \approx x$ for $x \in [0,1]$? The hint provided was to rewrite $x = \frac{1}{2}+\Delta$, assume $w_1$ is ...
2
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1answer
46 views

Find $\lim\limits_{x\to 0}(\cos(xe^x)-\ln(1-x)-x)^{1/x^3}$

Find $$\lim\limits_{x\to 0}(\cos(xe^x)-\ln(1-x)-x)^{1/x^3}$$ $$\lim\limits_{x\to 0}(\cos(xe^x)-\ln(1-x)-x)^{1/x^3}=e^{\lim\limits_{x\to 0}\frac{\ln(\cos(xe^x)-\ln(1-x)-x)}{x^3}}$$ Using Taylor ...
5
votes
3answers
228 views

Poisson Process - non-zero probability of more than one arrival

Quoting Bertsekas' Introduction to Probability: An arrival process is called a Poisson process with rate $\lambda$ if it has the following properties: a) Time homogenity - the probability $...
1
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1answer
55 views

Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
1
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2answers
35 views

Derive Taylor Series for $f(x)$

I am learning to derive the Taylor Series for $f(x)$, and I cannot remember how to do the following integral. $\int_{x_0}^x \left(x-x_0\right) \, dx $ to get the following solution $=\frac{\left(x-...
0
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0answers
40 views

Different forms of remainder in Taylor series

In the literature, it is common to find different forms of the remainder function in a truncated Taylor series. To name a few: Integral form Lagrange form Cauchy form First, can you tell me any ...
4
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3answers
138 views

What is the 90th derivative of $\cos(x^5)$ where x = 0?

Trying to figure out how to calculate the 90th derivative of $\cos(x^5)$ evaluated at 0. This is what I tried, but I guess I must have done something wrong or am not understanding something ...
1
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2answers
37 views

Using Maclaurin series, finding the value of an infinite sum…

I want to find $$\sum\limits_{n = 1}^\infty {{{{{({1 \over 2})}^n}} \over {n(n + 1)}}} $$ The book that has this problem in says to use the Maclaurin series for $(1 - x)\log (1 - x)$. I don't ...
1
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1answer
94 views

Maclaurin's Series for $\sec(x)$ with help of Maclaurin's series for $\tan(x)$

Is there any way to derive Maclaurin's series for $\sec(x)$ with the help of Maclaurin's series for $\tan(x)$? As we know, the Maclaurin's Series for $\tan(x)$ is: $$\tan(x)=x+\frac{x^3}{3}+\frac{...
0
votes
1answer
22 views

Expansion of $x \log \left(\frac{l+x}{x} \right)$ about x=0

I've read that $$x \log \left(\frac{l+x}{x} \right)=x \log \frac{l}{x} + O(x^2).$$ I tried to derive this using the usual Taylor series method but kept getting a division by zero. Could anyone ...
0
votes
1answer
27 views

Maclurin series for $\sin^2(x)$

I am trying to find the maclurin series expansion for $\sin^{2}x$. First I used the half angle identity: $$\frac{1-\cos(2x)}{2}=\sin^{2}x$$ Then substituted in the maclurin series for $\cos(2x)$ to ...