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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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
31 views

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

$$x^2+3e^x=x^2+3\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}+O(x^n)=x^2+3\left(1+x+\frac{x^2}{2}+O\left(x^2\right)\right)$$ ...
3
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3answers
65 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 ...
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4answers
50 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} ...
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2answers
70 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 ...
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} - ...
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0answers
22 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 ...
0
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1answer
47 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 ...
2
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2answers
27 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
16 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 ...
0
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1answer
44 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 $\ln(x)$ ...
1
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1answer
26 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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0answers
24 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 ...
1
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1answer
65 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 ...
2
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0answers
49 views

Solving for a $v$ in $\sum a_i e^{b_i (z^2+d_i) + c_i v}$

I have an equation in complex domain, $$P(e^u,e^v)=\sum_{i=1}^{N} a_i e^{b_i u + c_i v}=0 \;\;\;\text{(A)}$$ and by redefining, at the roots (I'm only showing work for one root), the first ...
0
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1answer
39 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: ...
4
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1answer
170 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 ...
2
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0answers
38 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 ...
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1answer
16 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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44 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 ...
3
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2answers
99 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 ...
1
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2answers
27 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 ...
1
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1answer
17 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) ...
1
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1answer
40 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 ...
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4answers
229 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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2answers
43 views

Find the series expansion of $\text{csch}^{-1}(x)$

Find the series expansion of $\text{csch}^{-1}(x)$ $\text{csch}^{-1}(x)=\ln 2-\ln x+\frac{x^2}{4}-\frac{3}{32}x^4+\frac{5}{96}x^6-...$ $\text{csch}^{-1}(x)=\ln(\sqrt{1+\frac{1}{x^2}}+\frac{1}{x})$ ...
8
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3answers
78 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 ...
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1answer
56 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: ...
1
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1answer
43 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 ...
0
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0answers
23 views

Taylor and Laurent equal in analytic domain: $\sum_{n=0}^\infty\frac{F^n(0)}{n!}z^{-n}=\sum_{n=0}^\infty\frac{F^n(0)}{n!}z^{-n}$?

If I have a taylor series around zero that looks like this: $\sum_{n=0}^\infty\frac{F^n(0)}{n!}z^{-n}$ Can I claim that this is equal to the first half of the Laurent series: ...
2
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1answer
34 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
44 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 ...
0
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0answers
32 views

What is the Lagrange remainder in a Taylor series expansion

I know what a Taylor series expansion is and I know how to find the Lagrange remainder but what does it mean intuitively? I need an explanation of what the Lagrange remainder represents in terms of ...
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0answers
20 views

Fluid Flow and Uniform Convergence of Taylor Series

I'm reading through a text on fluid flow and Laplace's equation, and it makes a statement that I do not understand and would really like to clarify. Here's the setup: Let $u:A \rightarrow R^2$ a ...
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2answers
33 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 ...
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0answers
33 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
129 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 ...
2
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2answers
34 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 ...
0
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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
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1answer
26 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 ...
1
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1answer
47 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: ...
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1answer
28 views

Taylor Series Exercise

What method should I use to find the Taylor series of $f(x)=\frac{x+2}{2-3x}$ with center 2? Here's what I did: Let $y=x-2$ ...
2
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1answer
51 views

Real analytic way to explain why the radius of convergence of $1/(1+x^2)$ is small

For any series expansion of $\frac{1}{1+x^2}$, the disc of convergence is blocked by the two singularities on $+i$ and $-i$. A series expansion about $0$ gives a radius of convergence of $1$. Is ...
0
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1answer
68 views

Finding a function that is harmonic in an annulus,

The problem statement is: Suppose that the real series $∑_0^{∞} a_n$ and $∑_0^{∞} b_n$ converge absolutely. Part 1 Prove that there is a function $u(r,θ)$ which is harmonic in $1<r<2$ and ...
1
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1answer
29 views

Taylor expanding $f(y+\epsilon U1 + {\epsilon}^{2} U2,t,\epsilon)$ in $\epsilon$

How would one Taylor expand $\epsilon f(y+\epsilon U1 + {\epsilon}^{2} U2,t,\epsilon)$ in $\epsilon$? Somehow the professor obtained the first few terms to be: $\epsilon f(y+\epsilon U1 + ...
2
votes
2answers
36 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 $0.0001$?
4
votes
1answer
41 views

How to find the Maclaurin series for $f(x) = \frac{1}{1 + \sin(x)}$?

I have that $\frac{1}{1 + x} = 1 - x + x^2 - x^3 + ...$ So then $\frac{1}{1 + \sin(x)}$ should be $ 1 - \sin(x) + \sin^2(x) - \sin^3(x) + ...$ but clearly this is not the case. So how does ...
1
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1answer
37 views

complex series expansion for $f(z)=\frac{1}{z-1}$

Expand the function $f(z)=\frac{1}{z-1}$ as as a series around $z_{0}$ in two regions a) $$|z-z_{0}| < |1-z_{0}|$$ b) $$|z-z_{0}| > |1-z_{0}|$$ and find coefficient $a_{n}$ is each case. I ...
0
votes
2answers
47 views

Power Series Approximation of ln(x)

I am working on building a small embedded calculator, and am working on adding a natural logarithm function that utilizes only + and -. I have worked out the power series representation of ln(x) as ...
0
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1answer
26 views

Why can the first term of the Taylor series expansion of $\cos(\theta)$ be written in the way below?

Why can the first term of the Taylor series expansion of $\cos(\theta)$ be written as $\cos(\theta_0) - (\theta - \theta_0) \sin (\theta_0)$?