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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25 views

Taylor polynomial, Peano form of the remainder of f(x) and its asymptote

Could someone help me complete this or check if my reasoning so far is correct? I'm stuck at finding the oblique asymptote: Write the Taylor polynomial and the Peano form of the remainder of $f(x) = ...
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2answers
38 views

Does $\sum\limits_{n=1}^\infty \left[n\left(f\left(\frac{1}{n}\right)-f\left(-\frac{1}{n}\right)\right)-2f'(0)\right]$ converge?

Let $f\in C^3([-1,1])$ Is the series $\sum\limits_{n=1}^\infty \left[n\left(f\left(\frac{1}{n}\right)-f\left(-\frac{1}{n}\right)\right)-2f'(0)\right]$ convergent? I'm trying to use Taylor's ...
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3answers
58 views

standard Taylor series using substitution

Find Taylor series using substitution about $0$ for $f(x)=\frac{125}{(5+4x)^3}$ by writing $\frac{125}{(5+4x)^3}=\frac{1}{(1+\frac{4}{5}x)^3}$? Determine a range of validity for this series.
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2answers
33 views

How to find the exponent $z$ of $(-1)^z$ for a patterned series of signed ones?

A question in Larson "Calculus" asks for the Taylor series centered at $\frac{\pi}{4}$ of $\cos(x)$. This expands to: $a_n = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) - ...
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3answers
112 views

A system of $n$ equations , how does it behave for growing $n$?

I read about the system of $n$ equations in the link below. I wonder how it behaves for growing $n$. Does it converge ? http://math.eretrandre.org/tetrationforum/showthread.php?tid=889 Here it is ...
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2answers
107 views

Taylor's Theorem Problem

This is from my engineering mathematics textbook. Is this version of taylor's theorem correct ? Successive Differentiation, Maclaurin's and Taylor's Expansion of Function $-147$ TAYLOR'S THEOREM ...
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0answers
26 views

Entire periodic $f(z)$ with more than 50 % of the derivatives $0$?

Im looking for a real-entire function $f(z)$ such that for any complex $z$ : $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number. $2)$ $f(z)= 0 + a_1 z + a_2 z^2 + a_3 z^3 + ...$ where more than ...
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27 views

Looking for a family of entire functions.

Let $i$ be a positive integer and let $f_i(x)$ be the $i$ th real transcendental entire function that has the Taylor expansion $f_i (x) = a_{i_0} + a_{i_1} x + a_{i_2} x^2 + ...$ with all $a_{i_n}$ ...
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21 views

Basis for a general function of tr$X^k$

I have a function of a $4\times 4$-matrix variable $X$ which is a general function of $\left<X^k\right>$ where $\left<\cdot\right>$ denotes the trace. My question is this: is there a ...
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120 views

Prove that $\lim_{x\to\infty} f'(x) = 0$ [duplicate]

Let $f(x)$ be twice differentiable on $(0,\infty)$ and let $\lim_{x\to \infty} f(x) = L<\infty$ and $|f''(x)| \le M$ for some $M>0$. Prove that $\lim_{x \to \infty} f'(x) = 0$. I've tried to ...
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0answers
24 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
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1answer
29 views

Taylor's series and the function argument dimension

I've stumbled over an interesting question. In $\cos(x)$, $x$ is measured in, say, radians. When I expand cosine in Taylor's series, I have the terms with $x^3$, $x^5$ etc. so I am summing up ...
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1answer
29 views

Geometric interpretation of a Taylor series like identity

Johann Bernoulli published (something like) the following expression in his journal Acta Eruditorum. $\int_0^x f(t) dt = xf(x)-\frac{x^2}{2!}f'(x)+\frac{x^3}{3!}f''(x)-\frac{x^4}{4!}f'''(x)+...$ Is ...
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1answer
316 views

Taylor series for cosine around $\pi/3$

I need the Taylor-Series for $ f(x) = \cos(x) $ in $ a = \pi/3$: \begin{align*} f(x) &= \cos(x - \pi/3 + \pi/3) \\ &= \cos \left( x - \frac{\pi}{3}\right) \cos\left(\frac{\pi}{3}\right) - ...
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24 views

$X\sim\mathcal N(0,1)$, Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd?

If $X\sim\mathcal N(0,1)$ Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd ? ($\Phi_X^{(j)}(0):j^{th}$ derivative of the characteristic function of the r.v. $X$) We computed ...
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5answers
949 views

What's the Maclaurin Series of $f(x)=\frac{1}{(1-x)^2}$?

This function seemed to be pretty much straight forward, but my solution is incorrect. I have two questions: 1. Where did I make a mistake? 2. I learned that there are shortcuts for finding a series ...
2
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1answer
24 views

Approximating error using Taylors theorem

I have used a Maclaurin series for the function $f(x) = \cos(2x)$ and have successfully produced: $\dfrac{2^n cos(\frac{n\pi}2)x^n}{n!}$ Now I want to estimate the error in approximating $\cos(2x)$ ...
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1answer
222 views

Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?

Im looking for a real-analytic function $f(z)$ such that for any $z$ $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number and where $z$ is close to , or onto the real line such that $z$ is in the ...
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3answers
28 views

Maclaurin series stuck at finding $L_n$

I need to develop Maclaurin serie of $f(x)=\frac{1}{(1-x)^2}$ I found all the derivative, and all the zero values for the derivatives. I come up with that : ...
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1answer
49 views

Let $f:[-1,1] \to \mathbb{R}$ be differentiable 3 times, prove $\exists M>0 \ , \ s.t \ f(x) \le Mx^2$

Let $f:[-1,1] \to \mathbb{R}$ be differentiable 3 times, let $f(0)=0$ and $f(x) \ge 0 \ \forall x \in [-1,1]$. Prove: $\exists M>0 \ , \ s.t \ f(x) \le Mx^2$. I separated the proofs to ...
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2answers
24 views

How to show MacLaurin series for $\frac{1}{1-x}$ converges using remainder term

$\dfrac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots$ this is valid for $x$ between $-1$ and $1$ not including the endpoints. How can one show that the Lagrange remainder goes to zero as $n$ goes to ...
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1answer
14 views

First Order Approximation Taylor Series

I have the taylor series $f(z)=f(x_0)+(x-x_0)f'(z)+1/2(x-x_0)^2f''(z) ...$ and I am told that "As a first order approximation," $x-x_0$ ~ $\frac{f(x)-f(x_0)}{f'(x_0)}$ assuming $f'(x_0) \neq 0$ I ...
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1answer
43 views

Finding Laurent series where given annulus is not in a singularity

I'm given a problem where I need to calculate the Laurent series of $f(z)$ inside the given annulus $$ f(z) = {1\over z^3(z-1)}; \quad 1 < |z| < 2 $$ From online resources(videos, notes) I ...
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1answer
50 views

Maclaurin series of (1+x)^(1/x)

how can i find the Maclaurin series of $f(x)=(1+x)^{1 \over x}$? $f(0)$ is not even defined, or should I define it as $f(0)=e$? I stopped at the first derivative as it gets terribly messy. thank ...
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1answer
23 views

Using Taylor's series in imporper integrals

Is it possible to simplify an improper integral using Taylor's series? How can I prove this procedure is correct? For example, take $$f(\alpha)=\int_0^{\infty} ...
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2answers
60 views

The simplest way to pow using only simple arithmetic

i want to get function $f(x, a) = x^a$, for both x and a - real numbers, that uses only + - * /. So only way I found is: get taylor series for $$x^a = \sum_{n = ...
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2answers
3k views

Taylor series expansion for $f(x)=\sqrt{x}$ for $a=1$

I seem to be stuck defining an alternating sequence of terms in this series because $f^{(0)}(x)=f(x)$ is positive, as well as $f'(x)$, but then every other term starting with $f''(x)$ is negative. How ...
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1answer
96 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
10 views

Taylors formula

I have a circle $K(a,\epsilon) \subset \Omega $ and for $ \parallel \Delta x \parallel \lt \epsilon $ we look at $ \Delta f = f(a+ \Delta x) - f(a) $ Now I look at the function $ F:[0,1]\rightarrow ...
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1answer
36 views

Taylor series $\ln(2+x)$ centered at $x=2$

Taylor series $\ln(2+x)$ centered at $x=2$. Is the correct result $$y=\ln \left(4\right)+\sum _{n=1}^{∞}\frac{\left(-1\right)^n}{4^{\left(2^{\Large n}\right)}}\cdot \frac{\left(x-2\right)^n}{n!}\ ?$$ ...
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2answers
30 views

Meaning of interval of convergence when approximating functions

Let's say I have a Taylor series approximation, $p(x)$, of a function $f(x)$ at $a$: $$ p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n} $$ And that this Taylor series has a radius of ...
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2answers
119 views

Proving $\forall x\in\mathbb R : \dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$ with Cauchy's MVT

Prove for all $x\in\mathbb R$: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ Mclauren expansion: $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+R_4(x)$$ ...
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0answers
30 views

Taylor expansions, inequalities and more

(Part A) I have to find the Taylor expansion of order 2 around (0,0) of $$f: \mathbb{R}^{2}\rightarrow \mathbb{R}$$ $$(x,y)x \mapsto f(x,y) = x\log (1+y)+sin(x+y) $$ Furthermore I have to prove if ...
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1answer
55 views

Finding $\zeta(4)$ by Taylor series

Is it possible to solve Zeta(4) function using something similar to the solution for zeta(2) as seen in this video? https://www.youtube.com/watch?v=mTPKyC3Udns
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20 views

solution check for approximating derivative using a Taylor expansion.

I'm wondering if there's in a mistake in either my reasoning or the given solution for the problem and was hoping to have someone double check this for me. The problem states: Let $g(2)=3$ , ...
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3answers
262 views

Why is domain of convergence of Taylor series of $\ln(x)$ about $x=1$ is $ (0,2)$?

I can understand the lower bound as $\ln(x)$ doesn't exist for $x<0$. But how is the upper bound $2$?
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1answer
16 views

taylor series please i need immediate help

Expand 1/z by Taylor series about a point z=1. what I have done really makes no sense because I have no idea about it. I can only think of 1/z=1/z-1+1 1/z=1/((z-1)*(1/z-1))
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3answers
62 views

When does the remainder term in the taylor series go to zero?

When does the remainder term in the taylor series go to zero? Theorem: Let $f\in C^{N+1}([\alpha,\beta])$ and $x,x_0\in(\alpha,\beta)$. Then ...
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4answers
77 views

If $\displaystyle \sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n= 0$ for any $n$ [closed]

Suppose that $f(x)=\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}$ for all $x$ with the radius of convergence $R>0$. If $\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n=0$ for any $n$.
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Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?

Using the Taylor expansion $$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain $$f(x+a) = ...
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2answers
52 views

Taylor series of $\sin(x)$ converges uniformly on $[-\pi,\pi]$?

According to my notes, the Taylor series of $\sin(x)$ converges uniformly on $[-\pi,\pi]$. I know that the remainder term needs to converge uniformly to $0$ for this to be the case. But I really ...
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1answer
51 views

How can I find the Maclaurin series of $\frac{1}{1+x+x^2}$?

How can I find the Maclaurin series of $\frac{1}{1+x+x^2}$? At first, I found the Maclaurin series of $\frac{1}{1+x}$, which is $\sum_{n=0}^{\infty}(-1)^{n}x^{n}$ and simply replaced $x$ with $x^2 + x ...
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2answers
55 views

$f(x)=\begin{cases}e^{\frac{-1}{x^2}} & \text{ if } x\neq 0 \\ 0& \text{ if } x= 0\end{cases}$ is not equal to its Maclaurin Series

$f(x)=\begin{cases}e^{\frac{-1}{x^2}} & \text{ if } x\neq 0 \\ 0& \text{ if } x= 0\end{cases}$ is not equal to its Maclaurin Series, which is ...
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1answer
28 views

Expanding $\frac{1}{\sqrt{1-x^2}}$ through the expansion of $\frac{1}{\sqrt{1-x}}$ by binomial series

I heard that to expand $\frac{1}{\sqrt{1-x^2}}$, I have to expand $\frac{1}{\sqrt{1-x}}$ by binomial series and then just replace $x$ to $x^2$. Using binomial series, I found that ...
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4answers
125 views

Two ways to show that $\sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$

Show that: $\large \sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ on: $0<x<\frac {\pi}2$ I tried to solve it in two ways and got a little stuck: One way is to use Cauchy's MVT, define $f,g$ ...
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3answers
30 views

How can I expand $\frac{1}{\sqrt{1-x^2}}$ by using the binomial series?

How can I expand $\frac{1}{\sqrt{1-x^2}}$ by using the binomial series? I know how to expand $\frac{1}{\sqrt{1-x}}$, but I have no idea how to expand $\frac{1}{\sqrt{1-x^2}}$. Simply differentiate ...
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1answer
38 views

Cubic MacLaurin $e^{x^2}$

Find the Cubic MacLurin expansion of e^{x^2}. First, I tried the sub $t=x^2$ and used the regular expansion for $e^t$. But that was wrong. Can I not do non-linear substituions? My calculations: ...
0
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0answers
43 views

Understanding Taylor's Theorem

In our real analysis course, our lecturer has given us the following theorem, which I don't quite understand. It's been given in an odd way, not similar to anything I've found in books or on the ...
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2answers
39 views

How to show that $\cos(x)=\sum\limits_{n=0}^\infty (-1)^n \frac {x^{2n}}{ (2n)!} $

We know that cos(x) is infinitely differentiable and the Lagrange remainder $\rightarrow 0$ for all $x$, so the Taylor series indeed produces the function. We also know that $\cos^{(4k)}(x)=\cos(x)$ ...
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1answer
16 views

How can I find the Taylor series of a function using the known Taylor series of a related function?

I am trying to calculate the Taylor series for the function: $$f(x) = {\frac 1 x}(1 - \cos\sqrt{x})$$ How do I do it, if I know the Taylor series for $\cos(x)$? $\cos x = {\Large \sum\limits_{k = ...