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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185 views

How can I prove that $e^x \cdot e^{-x}=1$ using Taylor series?

When proving $e^x.e^{-x}=1$ by using Taylor series, there are infinite many terms of $e^x$ and $e^{-x}$. Is there any fancy way to combine terms by terms to show that eventually it is equal to $1$?
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0answers
16 views

what's the taylor serie and it's convergence

I have this problem: What is the Taylor series of $\sqrt{x}$ at $x_0 = 4$. What is its interval of convergence? I am stuck and I can not finish it. Any idea on how to do that? Thank you
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0answers
16 views

Questions about the proof of Quadratic convergence with taylor's theorem

First of all, why do we take the absolute value of both sides? What is the point/reason? and for (x_n - z)^2 , isnt it always positive? Second of all, do you call this a recurrence relation? for the ...
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2answers
17 views

Hessian at a non-stationary point

I have a function $G(Q) : \mathbb{R}^n \rightarrow \mathbb{R}$ that is known to be convex. I also know that $Q^*$ is a minimum of $G(D)$. If I apply Taylor's theorem to $G(Q)$ at $Q^*$, I get: $$ G(Q)...
2
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2answers
106 views

Branch cut for $\sqrt{1-z^{2}}$ and Taylor's expansion!

I'm working in a problem that involves the equation $$ w(z)=\sqrt{1-z^{2}} \,\, . $$ I already know that there're two branch points in this equation, namely $\pm 1$, so there's a Riemann surface ...
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0answers
27 views

Is the proof of the statement make sense?

Please refer this link for some background material http://www.docdroid.net/161p6/curve.pdf.html So i propose a statement to a online tutor, the answer at the below link is the proof of the ...
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1answer
65 views

Is my idea of decomposing a meromorphic function into a sum of Laurent series correct?

We know that complex-analytic functions $f(z)$ agree with their power series representations on their domain of analyticity. If a meromorphic function has simple poles at $z_1, ..., z_m$ and $\...
2
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1answer
46 views

Counter example to theorem in complex domain

A theorem on Taylor series in complex domain is as follow: Suppose $f(z)$ has Taylor series at $a$ with convergence radius of $R$. Then $f(z)$ has at least one singular point on $|z-a|=R$. But I ...
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3answers
38 views

Taylor expansion of a complex function on a disc

I need to find the taylor expansion of the complex function $\frac{z^2}{z-2}$ on the disc $|z|<2$ I'm not sure how to start this off, can anyone help me?
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2answers
83 views

Why does each successive term in a Taylor series need to be much less than the previous term?

This is an extension to this previous question for this original question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works ...
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2answers
64 views

Let $\mathbf A$ be a matrix such that $\mathbf A^2=-\mathbf I$. Prove that $\exp(\varphi\mathbf A)=\mathbf I\cos{\varphi}+\mathbf A\sin{\varphi}$

Let $\mathbf A$ be a matrix such that $\mathbf A^2=-\mathbf I$. Prove that $\exp(\varphi\mathbf A)=\mathbf I\cos{\varphi}+\mathbf A\sin{\varphi}$ This is my attempt: $$\mathbf A^2=-\mathbf I \...
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0answers
38 views

Series expansion of $ (1-\frac{1}{x})^x $ at infinity

I'm trying to compute the expansion of $ (1-\frac{1}{x})^x $ at infinity, which is given by WolframAlpha as $$ \frac{1}{e} - \frac{1}{2ex} - \frac{5}{24ex^2} - \frac{5}{48ex^3} - \frac{337}{5760ex^4} +...
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0answers
24 views

Product with multi-dimensional matrix

We know the power series in scalar case $f:\mathbb R\to\mathbb R$ $$ f(x)=a_0x^0+a_1x^1+a_2x^2+a_3x^3+\dots $$ so what is the extension to the multi-dimensional case $f:\mathbb R^n\to\mathbb R$ $$ f(x)...
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2answers
35 views

How many n derivatives do you take for Taylor series to be accurate? [closed]

How many derivatives must we take to consider some Taylor series an accurate reflection of a function?
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1answer
53 views

Solution of the functional equation $g(x)g(z) = g(x+z)+g(x-z)$

What is the solution for the following functional equation? $g(x)g(z) = g(x+z)+g(x-z)$ The solution given is: $g(z) = 2\cos(z)$. In the derivation of the result (using Taylor expansion), there is ...
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1answer
54 views

Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
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1answer
21 views

Coefficients of power series

After expansion, we have $$ (x_1+x_2+\dots+x_n)^m=a_1x_1^m+a_2x_1^{m-1}x_2+\dots $$ where $x_{()}$ is the variable and constant indices $n>m$. What is the expressions of all these possible ...
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0answers
50 views

Why is the Taylor series of $1/\sqrt{1-4q^2}$ popping up in my recursively defined triangle of polynomials?

While answering this question I stumbled on some nice (inexplicable) observation where a recursively defined sequence of polynomials turned out to coincide with some Taylor development I'll start ...
2
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1answer
54 views

How do you Taylor expand the log likelihood function of the Poisson distribution?

This question is an extension to this previous question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works through the case of a ...
2
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2answers
25 views

Prove the series expansion

Prove that $$(1+x)^\frac{1}{x}=e-\frac{e}{2}x+\frac{11e}{24}x^2-\frac{7e}{16}x^3....$$ where e is exponenial , can any one give a proof...I tried with series expansion i could not get it.
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3answers
40 views

Prove the following using Maclaurin's theorem

Prove that $$\log(1+e^x)=\log 2+\frac{1}{2}x+\frac{1}{8}x^2-\frac{1}{192}x^4......$$ I have tried doing it. Tell me if you think the question is wrong
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3answers
89 views

How to show $1 +x + x^2/2! + \dots+ x^{2n}/(2n)!$ is positive for $x\in\Bbb{R}$?

How to show $1 + x + \frac{x^2}{2!} + \dots+ \frac{x^{2n}}{(2n)!}$ is positive for $x\in\Bbb{R}$? I realize that it's a part of the Taylor Series expansion of $e^x$ but can't proceed with this ...
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2answers
49 views

Taylor series of $\ln(x+2)$

I try to determine the Taylor series of $\ln(x+2)$. Since I know $\ln(1-x) = \sum_{n=1}^{\infty} \frac{x^n}{n}$, I suppose I can rewrite, \begin{align} \ln(x+2) &= \ln(1-(-(x+1)))=-\sum_{n=1}^{\...
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1answer
97 views

Taylor series and radius of convergence: $\sqrt{x}$ with centre $x = 16$?

I've been struggling with this question for a while now and getting nowhere with it. Could someone please help me out? Assuming that the function has a power series expansion about the given point, ...
3
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0answers
47 views

Which version of Taylor Theorem is this?

Suppose $X$ is a random variable and $\psi(t)=E[\exp(itX)]$ is its characteristic function. Let $K(t)$ be the principal value of the logarithm of $\psi(t)$. Suppose further that $E(|X|^{r+2})<\...
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0answers
42 views

Taylor polynomial approximation with absolute error less than 3 decimal places

Assume that f is a function with $|f^{(n)}(x)| \le 11,$ for all n and all real x. Let $T_n(x)$ denote the Taylor polynomial of degree n for f(x) about the point $x=0$. What is the least integer n ...
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1answer
33 views

Finding the error in a two-step finite difference numerical approximation

I got the following question in a math lecture the other day, and I'm not really sure how to go about it: A differential equation is given in the form $$\frac{\partial y}{\partial x} = f (x, y(x)...
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4answers
102 views

Maclaurin polynomial of tan(x)

The method used to find the Maclaurin polynomial of sin(x), cos(x), and $e^x$ requires finding several derivatives of the function. However, you can only take a couple derivatives of tan(x) before it ...
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0answers
9 views

Taylor Remainder over an interval for polynomial interpolation

When attempting to find how big n should be so that $|e^x - p(x)| < 10^{-4}$ over the interval $[-1,1]$ using Taylor Remainder, what value should I be using for $x$ in $(x - x0)^{n+1}$? I'm using 1,...
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1answer
40 views

Why Taylor series “is convergent” to differential when $\Delta x$, $\Delta y$ go to $0$?

Let $f(x,y)$ be a smooth function. Let $\Delta x$ and $\Delta y$ denote small differences in arguments $x$ and $y$, respectively. For any $x_0,y_0$ we can find Taylor series centered at that point: $...
3
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1answer
61 views

Evaluation or asymptotic for $\int_1^x y\sin\left(\frac{2\pi (y-1) x}{y}\right)dy$

Truly, my genuine problem (see Appendix for context) is compute in a closed form or an asymptotic, for real $x\geq 1$, for $$\int_1^x\left(\int_0^{y-1}\cos\left(\frac{2\pi t x}{y}\right)dt\right)...
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1answer
29 views

Error on Taylor formula argument

Question: My solution: $$f''(x) = \frac{f(x+h) - 2f(x) + f(x-h)}{h^2} $$ $$f''(x) = \frac{1}h \frac{f(x+h) - 2f(x) + f(x-h)}h$$ $$f''(x) = \frac{1}{h} [f'(x)-f'(x) = 0]$$ So because the ...
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2answers
43 views

If the first nonzero derivative at $a$ is of odd order $n\ge 3$, then $a$ is a point of inflection

Statement to Prove: Let $f$ be a real valued function such that for a fixed point $a$ , $$f^k(a)=0;1\le k\le n-1;\\and\ \ f^n(a)\neq 0.$$ Then if $n$ is odd then $a$ is a point of inflection. ...
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1answer
16 views

Second degree multi variable taylor polynomial

Let f (x, y ) = x cos(πy ) − y sin(πx) point: 1,2 I am following the standard formula, which starts with taking the partial of f with regards to x twice, which gives me: ysin(πx)π But plugging in ...
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1answer
42 views

Upper Error Bound Taylor Series

(a) Given $f(x) = \sqrt{x}$, find its Taylor polynomial of degree 2 centered at $x=4$ and use it to estimate $\sqrt{5}$. (b) Use Taylor's theorem to give an upper error bound for the estimate in part ...
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2answers
32 views

Maclaurin $f(x)=\sin^4x,x\in R$

Write Maclaurin Polynomial$$T\small{10}(x)$$ for function $$f(x)=\sin^4x,x\in R$$ Maclaurin Polynomial: $$T10(x)=f(0)+f'(0)x+f''(0)\frac{x^2}{2!}+...+f^{10}(0)\frac{x^{10}}{10!}$$ For my problem ...
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2answers
37 views

The following is a Taylor Series evaluated a particular value of x, find the sum of the series.

This is the Taylor Series in question 1 + $\frac{2}{1!}$+$\frac{4}{2!}$+$\frac{8}{3!}$+...+$\frac{2^n}{n!}$+... I know how to find whether or not the series converges or diverges easily using the ...
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1answer
28 views

Evaluating integral using invalid substitution

I was trying to show that for suitable t: $$ 2\pi(1+t/(\sqrt{(1-t)(3-t)})=\sum_{0}^{\infty}(t^n\int_0^{2\pi}1/(2-cos(\theta))^nd\theta $$ By uniqueness this is clearly the Taylor series about $0$ ...
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1answer
41 views

Meaning of $C^k$ in Taylor's expansion [closed]

In the following statement, what does $f \in C^k$ mean? And why is there a $q$ for the last part of expansion? So now if I let $k = 2$, what does it mean? And will the expansion involve 3nd ...
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0answers
16 views

Remainder of Taylor approximation

Consider the ODE $\dot{x}=f(x)$ with $f(x)$ smooth and let $x_0$ be an equilibrium, i.e. $x(t)=x_0=\text{const}$ and $f(x_0)=0$. The substitution $x=x_0+y$ shifts the origin to $x_0$. With the new ...
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0answers
106 views

A funny question: Taylor polynomials and series associated with the Lost numbers $4, 8, 15, 16, 23, 42$

The interpolation polynomial for the "Lost" numbers $4, 8, 15, 16, 23, 42$ is $$ P(x)=60-\frac{612}{5}x+\frac{367}{4}x^{2}-\frac{235}{8}x^{3}+\frac{17}{4}x^{4}-\frac{9}{40}x^{5}. $$ That is, $P(1)=4$, ...
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1answer
88 views

Evaluate the improper integral $ \int_0^1 \frac{\ln(1+x)}{x}\,dx $

I am trying to evaluate $$ \int_0^1 \frac{\ln(1+x)}{x}\,dx $$ I started by using the Taylor series for $\ln (1+x)$ $$\begin{align*} \int_0^1 \frac{\ln(1+x)}{x}\,dx &= \int_0^1\frac{1}{x}\sum_{n=...
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5answers
64 views

Series expansion for $x$, when $x$ is small

Suppose that we are given the series expansion of $y$ in terms of $x$, where $|x|\ll 1$. For example, consider $$y=x+x^2+x^3+\cdots\qquad\qquad\qquad (1).$$ From this I would like to derive the series ...
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0answers
20 views

Will values assigned to divergent series match a taylor series past the radius of convergence?

With what I've seen in nearly every case this is true but there are some cases where the function goes to infinity. I'm thinking specifically $y=ln(x-1)$, $y=1/(x-1)$, and $y=(x-1)^2$ centered at 0 ...
0
votes
1answer
11 views

Taylor series for arctan without using knowledge of its derivative

I am trying to prove that $\frac{d}{dx}\tan^{-1}x = \frac{1}{1+x^2}$ specifically by using knowledge of the Taylor series of $\frac{1}{1+x^2}$, integrating term-by-term, and showing this is $\tan^{-1}...
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1answer
17 views

Can I integrate then differentiate this power series to derive the same result as the binomial series expansion?

I've tried something but I'm not getting the right answer, so I'm wondering why it doesn't work. I want to taylor expand $\frac1{z^2}$ about some point $a\in\mathbb{C}$. Here's what I did: \begin{...
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2answers
33 views

What could the ratio of two sides of a triangle possibly have to do with exponential functions?

Name says it all. The two seem so unrelated? What's more, if I'm not mistaken the exponential version contains an imaginary part. I'm kind of ignorant about imaginary numbers, but does this mean that ...
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1answer
43 views

How to perform taylor expansion with numerical differentiation formula

I am attempting to perform taylor expansion on the following numerical differentiation formula: $f'''(0) = \frac {−f(−3h/2) + 3f(−h/2) − 3f(h/2) + f(3h/2)) }{ h^3 }$ Over the reference interval [−3h/...
2
votes
0answers
79 views

Second order Taylor expansion of vector-valued function

I am wondering what is the second order Taylor expansion of a vector-valued function $f(x):\mathbb{R}^M\rightarrow \mathbb{R}^N$. I know that the gradient of a vector-valued function is a Jacobian ...
3
votes
5answers
148 views

Prove that $1+x+\frac{x^2}{2}+\dots+\frac{x^n}{n!}<e^x$ for all $x\in(0,\infty),n\in\mathbb{N}$

Intuitively this makes sense but I don't know how to formally show that this is true. I tried using induction but that got me nowhere .