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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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 ...
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1answer
44 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 ...
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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
39 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
87 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
46 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) &= ...
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1answer
52 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, ...
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46 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 ...
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28 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
24 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, ...
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4answers
68 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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7 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 ...
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1answer
38 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: ...
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1answer
59 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 ...
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1answer
25 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
33 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
12 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
31 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
29 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
25 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
39 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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15 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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102 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
79 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 &= ...
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5answers
61 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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19 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 ...
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1answer
10 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 ...
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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: ...
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2answers
30 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
39 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 ...
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38 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 ...
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5answers
145 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 .
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64 views

Am I pretty close to proving that e is irrational?

Show that $e=1+1/1!+1/2!+1/3!+…$ is an irrational number. Hint: show that, for all positive integers $p$, $0<p![e−(1+1/1!+…+1/p!)]<1$. Then conclude that $e$ cannot be a ratio of two integers ...
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1answer
32 views

Taylor's Theorem Question: finding $\lim\limits_{x \to 0} \frac{(x-\sin x)^{70}}{1-\cos (x^{105})}$

I am trying to calculate $$\lim\limits_{x \to 0} \frac{(x-\sin x)^{70}}{1-\cos (x^{105})}$$ Here is my attempt: $ $ write $\cos$ and $\sin$ as Taylor series, and plug back into the original ...
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27 views

Prove the Taylor polynomial $p(x)=\sum_{j=0}^{k-1}{1\over j!}f^{(j)}(x_0)(x-x_0)^j$ interpolates $f$ at $x_0, x_0,…, x_0$ ($k$ repetitions).

Prove the Taylor polynomial $p(x)=\sum_{j=0}^{k-1}{1\over j!}f^{(j)}(x_0)(x-x_0)^j$ interpolates $f$ at $x_0, x_0,..., x_0$ ($k$ repetitions). I'm not sure how to approach this. Any solutions or ...
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4answers
35 views

Taylor series of $f(x)=\int_0^1 \frac{1-e^{-sx}}{s}ds$

Taylor series of: $$f(x)=\int_0^1 \frac{1-e^{-sx}}{s}ds$$ at $x_0 = 0$. I've done: By fundamental theory of calculus: $$f'(x)=1-e^{-1x}$$ Which is clearly differentiable by e.g. $n$ times. What ...
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1answer
31 views

Taylor's Theorem expansion

I need to show that $f'(x) = (f(x-2h) - 4f(x-h)+3f(x)) / 2h +0(h^2)$ with Taylor series expansion of $f(x-h)$ and $f(x-2h)$. I got the expansions but I don't get the final answer correct, so I think ...
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1answer
25 views

Determine the function of $f(z)$: singularities and residue

Can anybody help me by explaining step by step how to solve this question? The function $f(z)$ has a double pole at $z=0$ with residue $2$ and a simple pole at $z=1$ also with residue $2$. It is also ...
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1answer
22 views

Showing that there is $\theta \in (0, 1)$ such that $\sin(x + y) = x + y − \frac{1}{2}(x^2 + 2xy + y^2 ) \sin(\theta(x + y))$

Let $x, y \in \Bbb R$. Show that there is $\theta \in (0, 1)$ such that $$\sin(x + y) = x + y − \frac{1}{2}(x^2 + 2xy + y^2 ) \sin(\theta(x + y))$$ It seems like I need to somehow use Taylor's ...
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3answers
57 views

Why, intuitively, does the Maclaurin series for $e^x$ but not $\ln(1+x)$ converge globally?

So we all know that, $\forall x\in\mathbb{R}$, $$e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!}$$ And that $$\ln (1+x) = \sum_{k=0}^{\infty} \frac{(-1)^{k-1}}{k}x^k$$ But that this only holds for ...
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1answer
85 views

Show that $e^x=1+x+\frac{x^2}{2!}+…+\frac{x^n}{n!}+R_{n+1}$

Show that $\qquad$ $\qquad$ $e^x=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}+R_{n+1}$ with $\qquad \qquad$ $0 \lt R_{n+1} \lt e^x \frac{x^{n+1}}{(n+1)!}$ if $0 \lt x$ and $\qquad \qquad$ $|R_{n+1}| \lt ...
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0answers
20 views

Is there a name for this “simplified” Volterra series?

Consider a nonlinear, time-invariant system of the following form: $g(t) = \left[h_1(t) \ast f(t)^1\right] + \left[h_2(t) \ast f(t)^2\right] + \left[h_3(t) \ast f(t)^3\right] + ...$ where $\ast$ ...
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2answers
42 views

Why is this function is $O(x^n)$

$$f(x) = \prod_{k=0}^{n} (1+kx)^{ (-1)^k \binom {n} {k} }$$ How to prove that Taylor expansion of this function at zero stars from 1 and then $c x^n$ (all intermidiat terms are zero)?
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0answers
17 views

Radius of convergence of Taylor series without computing the Taylor series

Find the radius of convergence of the Taylor series $f(z) = \frac{z^2}{cos(z)}$ about the point $z = \frac{\pi}{4}$ without computing the Taylor series. Not sure how to approach this question without ...
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0answers
14 views

Which end-points should I choose to form the interval when using the bisection method?

I have the following problem: I am calculating the value of $\log X$ using some iterative functions. With each iteration of the function, the value of $\log X$ gets more precise. One of them is ...
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0answers
13 views

Taylor series of $1+2x^2+3x^4+\sin(e^{2x}+1)\log(1+x^4)x^4$?

Taylor series of $1+2x^2+3x^4+\sin(e^{2x}+1)\log(1+x^4)x^4$? I'm wondering whether this can be gained by considering the individual terms or by differentiating the whole? The differentiation of this ...
1
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1answer
25 views

infinite summation of derivatives of a convergent function

How can I simplify this summation $$\sum_{i=1}^\infty \left[1-\sum_{n=0}^{i-1}(-1)^n \frac{a^n}{n!} \left. \frac{d^n}{dt^n} f(t)\right|_{t=a} \right] $$ if $f(t)$ is equal to ...
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1answer
29 views

System of recurrence relations with Taylor series expansion

Find $a_n,b_n$ where $a_0=1,b_0=0$ for the following relations: $a_{n+1}=2a_n+b_n$ $b_{n+1}=a_n+b_n$ Using generating functions, the system is: $f(x)-a_0=2xf(x)+xg(x)$ $g(x)-b_0=xf(x)+xg(x)$ ...
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0answers
48 views

is it true that $ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx - \frac{1}{1-\epsilon}x $?

Target is to approximate $\frac{1}{\epsilon}\ln (1- \epsilon x) $ ($\epsilon, x \in (0,1) $). Here is one using $\ln (1+y) \approx y $: $$ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx -x $$ I ...