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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computing maclaurin series for $(\sin x)^3$ , order $3$

I have a clarification to ask: I want to compute $f(x)=(\sin x)^3$ by maclaurin series, order $n=3$. I know that: $\sin x=x-\dfrac{x^3}{3!}+R_3(x)$. So can i say that: $\sin^3x=(\sin ...
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21 views

Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...
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92 views

proving that $g(x)=0$ has one real root

Given $g(x)=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^{2n+1}}{(2n+1)!}$, Need to prove that $g(x)=0$ has one real root. I thought to use the fact that $e^x<T_{2n}(x)$ for all $x<0$, ...
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24 views

Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and approximate $R_n < \frac{1}{10000}$

I am tasked with the following: Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and use the power series to approximate $\displaystyle \int_{0}^{0.5}3e^{-x^2/2}$ with error ...
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25 views

Find MacLaurin polynomial of integral

I have not the slightest idea how to begin with the following problem. My first thought is to integrate it before trying to find the MacLaurin polynomial, but I don't know if that is possible. Here is ...
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27 views

Help understanding a question

I know this probably isn't the best question to post as far as further use with others, but I literally have no where else to turn to for study assistance. My problem is as follows: Find $T_5(x)$: ...
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1answer
24 views

Approximating an integral with taylor series

I am working on the following homework problem: "Assume that $\sin(x)$ equals its Maclaurin series for all $x$. Use the Maclaurin series for $\sin(5x^2)$ to evaluate the integral ...
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44 views

Taylor expansion in terms of known Taylor series

Let $f(x)$ be a "nice" function with the following properties: $f$ is a real analytic, strictly increasing, odd and bounded function, i.e. $f(-x)=-f(x)$ and $-1<f(x)<1$. Further, let $f(x)>0, ...
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26 views

Finding an expression for the general term of a taylor series

I am working on a homework problem that asks the following: "Find an expression for the general term of each of the series below. Use $n$ as your index, and pick your general term so that the sum ...
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90 views

About sparse polynomial squares

Given $p\in\mathbb{Q}[x]$, we define the weigth of $p$ as: $$ W(p) = \#\{n\in\mathbb{N}: [x^n]\,p(x)\neq 0\} $$ i.e. as the number of non-zero terms. By playing a bit with the Taylor series of ...
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63 views

Limit and Taylor Series when non-differentiable

I am stuck on a problem similar to this one. Define $$f(\theta,y)=\frac{g(\theta y)}{\int_0^1 g(\theta x)dx}$$ with $g(0)=0$ and $\lim_{t \to 0}g'(t)=+\infty$. I am interested in $\lim_{\theta \to ...
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2answers
31 views

advice on using Taylor Series for function approximation

I've recently covered the Taylor Series in my studies and have read through several of the posts here which deal almost exclusively with specific problems and proofs but none seem to be answering a ...
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1answer
28 views

Calculus question - Derivative limit

Using Taylor how can I calculate $(\sin(x^3))^{\frac{1}{3}}$ up until $O(x^{13})$,and one more enquiry: Can the function be differentiated on the real axis and if yes what is it's derivative? $f(x) ...
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1answer
33 views

Trouble with series question from STEP past paper

I have answered all parts of this question but the last part. By using the identity, $\cot x - \tan x = 2\cot 2x$ ...
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29 views

Polynomial approximation for a function implies an approximation to its square root?

Assume $f:[0,1] \rightarrow \mathbb{R}$ satisfies $f(t)\geq 0, f(0)=0$ I am looking for a machinery, which given a polynomial approximation of $f$ of a certain degree, determines the highest order ...
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2answers
43 views

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n. from the series expansions of $\sin x$ and $\cos x$, I get that $\tan ...
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68 views

Taylor Series Expansion of $\frac{1}{\sin 2x}$ and $\frac{1}{1-e^{-x}}$

How do you find the Taylor expansions of the expressions: $\frac{1}{sin2x}$ and $\frac{1}{1-e^{-x}}$ I'm not sure what to do since all the terms are in the denominator. Thanks in advance for any ...
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2answers
66 views

Enlarging set where “weighted” MacLaurin series of $\frac{1}{1 - x}$ equals $\frac{1}{1 - x}$

Is it possible to select real values $a_{n, k}$ so that $$f(x) =\lim_{n \to \infty}\sum_{k = 0}^{n - 1} a_{n, k} x^k = \frac{1}{1 - x} $$ for all $x \in \mathbb{R} \setminus \{1\}$ ? Failing ...
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1answer
39 views

Expanding a function into a series

I am trying to follow a proof in QFT notes, however I am unable to follow this step - it's basically Laurent/Taylor expansion but I have very little experience with it. It's claimed that: ...
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133 views

Help with this limit?

I am trying to focus on the limits of functions with similar series expansions and I stumbled on this. ...
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1answer
56 views

finding the partial bell polynomial of $e^x$

$$ \left(e^{x+z} - e^x\right) = \sum_{n=1}^\infty \frac{z^n}{n!} \frac{d^n}{dx^n}[e^x] $$ $$ \left(e^{x+z}-e^x\right)^k = \sum_{n \geq k} Y^{\Delta}_{e^x}(n,k,x)z^n $$ Where: $$ Y^{\Delta}(n,k,x) = ...
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55 views

3Dimensional runge kutta and Euler method ( help to verify the idea and proposition)

I been discussing this idea with a tutor for sometime. However it turn out that the proof is not comprehensible.Can someone please help to verify the the proof for 3D Euler method and runge kutta ...
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9 views

How to compare order of magnitude?

In Fan and Li's paper "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties", they provided a proof to Theorem 1. The very last part of the proof is as follows. Some ...
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323 views

Can the following trick be expanded upon?

Main Question What is the expansion of $d^{1+\epsilon}?$ Background I noticed the following trick (sometimes more laborious) to directly differentiate $ f(x) $ twice without differentiating it even ...
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88 views

Show $ \lim_{n \to \infty} \frac{\ f(y_n)-f(x_n)}{ y_n-x_n} = f'(x_0)$ using Taylor

Let $f:[a, b]\to R$ differentiable at $a<x_0<b$. Using taylor series show that if $x_n \to x_0^-$ and $y_n \to x_0^+$ then $$ \lim_{n \to \infty} \frac{\ f(y_n)-f(x_n)}{ y_n-x_n} = f'(x_0)$$ ...
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1answer
21 views

Proving a Partial Derivative Equivalence Using Taylor Series Expansion?

I'm studying computer vision, and one of the problems in my book is to prove that $\partial f/ \partial x = f(x+1) - f(x)$ It's been a while since I've touched Taylor Series, and so I'm not sure of ...
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3answers
58 views

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^3$. Show the equivalence: $f^{(k)}(0) = 0$ for $k=0,1,2 \iff \lim_{x\to 0} \frac{f(x)}{x^3}$ exists

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^3$. Show the equivalence: $$f^{(k)}(0) = 0 \quad k=0,1,2 \iff \lim_{x\to 0} \frac{f(x)}{x^3} \text{, exists.}$$ Trying: Since $f \in C^3$, implies $f, f', ...
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60 views

What happens when I convert a Taylor series into an integral?

Suppose we have the Taylor series of an analytic function: $$f(x) = \sum_{k=0}^\infty \frac{1}{k!} a_k x^k$$ Then I decide to (kind of) turn it into an integral: $$g(x) = \int_0^\infty ...
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27 views

Evaluating MacLaurin polynomial of composite function

I want to evaluate the MacLaurin polynomial (Taylor polynomial around 0) $p$ of $f(x) = \sin(x^3)$ of order $11$ at $x=1$ and do this as efficient as possible (without much computation). When I just ...
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17 views

multivariable Taylor polinomials

I'm trying to find the Taylor series of \begin{equation*}e^{-(x^2+y^2)}\cos(xy) \textrm{ : up to 4'th order around } (0,0) \end{equation*} \begin{equation*}e^y\tan(x) \textrm{ : up to 3'rd order ...
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3answers
74 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
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132 views

Does $\tan (x)$ equal $\frac{-1}{x-\frac{\pi}{2}}+\frac{-1}{x+\frac{\pi}{2}}+\frac{-1}{x-\frac{3\pi}{2}}+\frac{-1}{x+\frac{3\pi}{2}}+…$?

I set my Year 12 students a question involving the sums of rational functions $\frac{1}{x-n}$. The graph of a sum of these functions looks an awful lot like a tan graph. This led me to ask: Does ...
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28 views

Use a calculator to compute the error $|e^x-T_2(x)|$ at $x=1.1$

I don't believe i have learned to solve for the error. Any help would be greatly appreciated. I have computed $T_2$ at $x=0.8$ $$T_2=e^.8+e^.8(x-.8)+e^.8/2(x-.8)^2 $$
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101 views

Computing $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$.

Compute $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$ with a precision (Accuracy? Error? What is the formal expression?) of 0.01. Attempt: First of all: $\ln(x+1)=\sum_{k=1}^{\infty}{(-1)^{k-1}x^k\over ...
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20 views

Condition on the limit of the upper incomplete gamma function

I am trying to find a lower bound on $q$ such that $$\Gamma(2p,qt)=\int_{qt}^{\infty}{x^{2p-1}e^{-x}\,dx}>0$$ for all $t>0,p>0$, and $q<0$. At first, I tried expanding $\Gamma(2p,qt)$ ...
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37 views

$\ln $ and Taylor Series Expansion (what went wrong)

Edited Problem I'm trying to express $\ln{(1-(\frac{N}{K})^{\frac{1}{4}})}$ in terms of $\ln N$, where $K$ is a constant and $1 \leq N \leq K$. This also implies $\frac{N}{K} \leq 1$. Anyone ...
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1answer
142 views

How to use Chebyshev Polynomials to approximate sin(x) and cos(x) within the interval [−π,π]? [closed]

I have approximated sin(x) and cos (x) using the Taylor Series (Maclaurin Series) with the following results How can I use Chebyshev Polynomials to approximate sin(x) and cos(x) within the ...
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24 views

Why is this true - easy question concerning asymptotics of exponential

Suppose $\lambda > 0$ is constant as $t \searrow 0$. In my lecture notes it is written that $\left(1+\sum_{k=1}^{\infty} \frac{(-\lambda t)^k}{k!}\right) \lambda t = \lambda t + o(t)$ and ...
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1answer
53 views

Comparing Two Multilinear Polynomials based on Multivariable Taylor Expansion

Given two linear functions $f(x)$ and $g(x)$ defined on real values, let's say that I want to show that $f(x) > g(x)$ for all real $x > t > 0$. According to the order-1 Taylor expansion at ...
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2answers
82 views

A Taylor series expansion of $e^{ix}$

In Probability Theory by Athreya and Lahiri, they give a very elegant proof of Central Limit Theorem (The Lindeberg one) wherein they use a lemma: For $x \in \mathbb{R}$ and $r \geq 1$, ...
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32 views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: ...
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32 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
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35 views

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence.

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence. My problem here is the Taylor series. Computing the few first derivative is possible, but I can't seem to ...
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1answer
89 views

If $f(0)=0$ and $f''\ge 0$, then $f(a+b)\ge f(a)+f(b)$

Given $\ f$ so $\ f''(x) \ge 0$ for every $\ x \ge 0$, also $\ f(0)=0$. Trying to show that if $\ a,b \ge 0 \Rightarrow f(a+b) \ge f(a) + f(b)$ Using Taylor I used $\ f(0)=0$ and got ...
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1answer
31 views

Given that $f(1) = f'(1) = 1$, use Taylor polynomials to show that $\lvert f(x) - x \rvert \leq A(x - 1)^2$

Given that $\ f$ has continuous second derivatives in$\ [0,2]$ and $\ f(1)=f'(1)=1$, I'm trying to prove that for every $\ x \in [0,2]$ exists an A so that: $$ |f(x)-x| \le A(x-1)^2 $$ The second ...
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82 views

A derivation of the Euler-Maclaurin formula?

The generating function for the Bernoulli numbers $B_n$ is $$\frac{x}{e^x-1}=\sum_{n=0}^\infty\frac{B_n}{n!}x^n$$ The sum of an infinite geometric series is $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ ...
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3answers
36 views

Proof:Taylor expansion of inverse trigonometric functions

I find it quite difficult to remember the Taylor expansion of inverse trigonometric functions.Actually in school we have been just taught the series (for finding limits in calculus without teaching us ...
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45 views

How to determine Taylor series expansion of function $f(x) = \frac{\cos(x)}{x}$ about $a=1$?

Given function is $f(x) = \frac{\cos(x)}{x}.$ $y = x - a , y = x - 1$. $x = y+1 , f(y) = \frac{\cos(y+1)}{y+1}$ How to get Taylor series expansion about $1$ of this function? If it was needed to ...
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3answers
97 views

Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$ [duplicate]

Determine the Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$. $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}} = \arcsin(x)\frac{1}{\sqrt{1-x^2}}$ It is known: (1.) ...
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1answer
41 views

Prove that for every $x$ in the area of $0$ exists: $\ln(1+x)=\sum_{1}^{\infty}\frac{(-1)^{n+1}x^n}{n}$

I need to prove 2 things: Prove that for every $x$ in a neighbourhood of $0$ exists: $\ln(1+x)=\sum_{1}^{\infty}\frac{(-1)^{n+1}x^n}{n}.$ What I did is that I calculated the derivatives of ...