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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6
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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 ...
0
votes
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}^{\...
0
votes
1answer
105 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
votes
0answers
49 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})<\...
1
vote
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 ...
0
votes
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)...
0
votes
4answers
107 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 ...
0
votes
0answers
10 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,...
1
vote
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
votes
1answer
62 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)...
1
vote
1answer
30 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 ...
0
votes
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. ...
0
votes
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 ...
0
votes
1answer
43 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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 $0$ ...
-1
votes
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 ...
0
votes
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 ...
2
votes
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, $P(1)=4$, ...
5
votes
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=...
2
votes
5answers
65 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 ...
0
votes
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}...
1
vote
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{...
1
vote
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 ...
0
votes
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
80 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 .
1
vote
0answers
69 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
votes
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 ...
0
votes
0answers
30 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 ...
3
votes
4answers
36 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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
4
votes
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(-...
2
votes
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 \...
3
votes
0answers
31 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 $\ast$ ...
1
vote
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)?
1
vote
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 ...
0
votes
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 ...
1
vote
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}\...
1
vote
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: $f(x)-a_0=2xf(x)+xg(x)$ $g(x)-b_0=xf(x)+xg(x)$ ...
0
votes
0answers
75 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
votes
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
votes
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
votes
2answers
45 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
votes
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 ...
0
votes
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. ...