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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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 ...
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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, $P(1)=4$, ...
2
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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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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
54 views

Log-linearizing $Y_t=\int_0^1 F(X_{it}) di$

I want to prove that log-linearizing the expression $Y_t=\int_0^1 F(X_{it}) di$ yields: $$Yy_t \approx F'(X)X\int_0^1 x_{it} di$$ Where: $\{X_{it}\}_{i \in (0,1)}$ is a continuum of strictly ...
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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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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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1answer
42 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 $\left(\dfrac{b}{t+b}\...
0
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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
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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
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5answers
147 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 .
2
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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
68 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 q/...
3
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1answer
45 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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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
37 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
27 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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3answers
63 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 $x\in(-...
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1answer
24 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 ...
3
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0answers
30 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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1answer
44 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
23 views

Taylor coefficients of a function

I'm having some trouble trying to prove the following: Prove that in the Taylor Polynomial of $\:f(x,y)= \sin(xy)$, centered in $(0,0)$ just the coefficients of order $4k-2$ are nonzero, for $k \in\{...
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0answers
16 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
14 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 ...
0
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0answers
73 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 ...
0
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1answer
49 views

Evaluate thus limit using series: $\lim_{x\to0} (\sin x-\tan x)/x^3$

Evaluate thus limit using series: $$\lim_{x\to0} \frac{\sin x-\tan x}{x^3}$$ I know the value of this limit is -1/2, and I also know the series expansion for $\sin x$ is $$x - \frac{x^3}{3!} + \frac{...
0
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1answer
30 views

Derivatives of characteristic function

Let $\phi$ be the characteristic function for random variable $X$. I know that if $E [|X|] < \infty$, then dominated convergence implies existence of the first derivative, and in particular, $\phi'(...
2
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1answer
78 views

Approximating polynomial of higher degree

Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values of ...
2
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2answers
43 views

Taylor Series and Differentiation with Sigma notation $f(x) = \frac{x}{(2-3x)^2}$

Use Term By Term Differentiation to Find the Taylor Series about $x$=3 for Give The Open Interval of Convergence and express as sigma notation $\sum A_n(x-3)^n$ $f(x) = \frac{x}{(2-3x)^2}$ So I ...
3
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1answer
47 views

Value of the limit without (or with, but giving rigorous arguments) using the Taylor expansion of sin

I'm trying to evaluate the limit as $N\to \infty.$ $$\frac{ \left(\dfrac{\sin \frac{1}{N}} {\frac{1}{N}}\right)^{N} -1 }{\frac{1}{N}}.$$ Note first that, using L'Hôpital, one can easily show that ...
4
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3answers
75 views

Expand in Taylor series $\frac{1}{1-\sin{x}}$

Expand in Taylor series $\frac{1}{1-\sin{x}}$ I have an idea that $\frac{1}{1-\sin{x}} = 1 + \sin {x} + \sin^2 {x} + \sin^3 {x} + \dots$ But I don't know what to do next. Every sine expands in ...
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1answer
4k views

Series expansion of square root function

Could I please know how to expand: $$\sqrt{4-3x^2}$$ I simplified it to $\sqrt{1-\frac34x^2}$ but the question is if I let $x = x^2$, what will the coefficient of, say $x^5$ or $x^{10}$ be in the ...
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0answers
44 views

Find smallest $k$ such that the given trigonometric functions are $O(x^k)$

I feel like I do not quite grasp the concept of Big O Notation. From my understanding, if $f(x) = O(g(x))$ then $f(x)$ is at most $g(x)$ multiplied by some constant C, which makes decent sense to me. ...
3
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1answer
37 views

Logarithmic expansion with cosines

I found the following expansion in this paper: $$\log\frac{|\boldsymbol{r}-\boldsymbol{r'}|}{L}=\log\frac{r_>}{L}-\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{r_<}{r_>}\right)^n\cos[n(\phi-\phi'...
1
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1answer
35 views

Does point's neighborhood have no local extremum?

I have polynomial of some limited degree: $f(x,y) = a_1 + a_2x + a_3y + a_4xy + a_5x^2 + a_6y^2 + a_7x^2y + \ldots$ There is a point $p_0=(x_0,y_0)$, which is NOT a local extreme NOR inflection ...
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2answers
57 views

Find a bound for approximating Taylor Series

I'm struggling to figure out how to find a bound on my error for this problem: Let T_{6}(x) be the Taylor polynomial of degree 6 based at a = 0 for the function f(x)=\cos(x). Suppose you approximate ...
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0answers
29 views

Order Taylor series prediction

It is easy to add the Taylor expansion, less easy to multiply and even less easy to compose. That said, the main problem lies not in the calculation itself in the prediction orders that need to be ...
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1answer
33 views

Accuracy of Runge-Kutta compared to Taylor expansion

Let's say I have an ODE of the form : $y'(x) = f(x,y(x))$. I've been told that using the Runge-Kutta method for solving this ODE is equivalent to using Taylor expansion if $f(x,y(x))$ is linear in $y$ ...
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0answers
13 views

Sommerfield Expansion Taylor Expansion

Ugh... I can't figure this out and I DO NOT understand why. So this has to do with the Sommerfield expansion of the Fermi function (wiki) (another reference) The issue I'm having is we are supposed ...
0
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1answer
73 views

Let $s(n,k)$ denote the signless Stirling numbers of the first kind. Prove that…

Let $s(n,k)$ denote the signless Stirling numbers of the first kind. Prove that: $$s(n,2) = (n-1)!(1 + \frac{1}{2} + \frac{1}{3} +...+ \frac{1}{n-1})$$ -I haven't dealt with Taylor series expansion ...
4
votes
1answer
39 views

Second order differential

Suppose I have a function $f = f(x,y,z)$. Then, the first order differential, or the linear approximation is $$ df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial ...
2
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0answers
31 views

Asymptotic behavior of the function $e^{- \lambda t^2}$ when $\lambda$ is small

I wish to prove that when $\lambda$ is taken to be very small $$ \left| e^{- \lambda t^2} - \sum_{n=0}^N \frac{(- \lambda t^2)^n}{n!} \right| = O(e^{-\frac{a}{\lambda}})$$ for some constant $a \in \...
0
votes
2answers
67 views

prove $(1+\ldots+o(x^{n-1}))^4=(1+\ldots+o(x^{n-1}))$

I would like to prove that : $$(1+\ldots+o(x^{n-1}))^4=(1+\ldots+o(x^{n-1}))$$ i took that from the picture below My Proof: note that $$(1+x)^a = 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-...
0
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1answer
43 views

Differentials squared - Divergence in general orthogonal curvilinear coordinates.

I was reading this document on how to get some common operators when dealing with general orthogonal curvilinear coordinates. I am interested in particular in equation (12). It basically defines three ...
0
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2answers
30 views

$\exists C>0$ such that $\frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2-Cx}$?

It is well-known and easy to check that for any real $x$ it holds $$ \frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2}. $$ [To show this, it is sufficient to write explicitly their Taylor series ...
1
vote
1answer
19 views

Two dimensional taylor expansion of arbitrary function

Consider the function dependent on the variables $N_t$ and $N_{t-1}$. Call the function $f$ so $f = f(N_t, N_{t-1})$. Now suppose we could write $N_t = N^*+n_t$ where $N^*$ is constant, and $n_t$ ...
0
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1answer
21 views

Show a function behaves as a harmonic oscillator

We have a function $V(x)$ (potential energy) with $x$ being some variable. This function has a minimum at a certain $x_0$. We assume that $V(x)$ is an analytic real function of $x$ around $x_0$. ...