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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71 views

Taylor series $\frac{\sin x}{x}$ convergence

I needed the Taylor series for $f(x) = \frac{\sin x}{x}$ in $a = 0$. I started with $ f(x) = \frac{1}{x} \cdot \sin(x) $, used the existing $sin$ Taylor series and multiplied by $\frac{1}{x}$: $$ ...
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1answer
42 views

Convergence of an analytic function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a smooth function. Let $R$ be the radius of convergence of the Taylor series centered at $a.$ For each $n \in \mathbb{N},$ let $M_n= \sup\{f^{n}(t) : t \in ...
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332 views

Compute the first five non-zero terms of the Taylor series about $a=4$ for $f(x)=\sqrt{x}$

This is my first Taylor Series problem and I want to make sure I completed it correctly. Here is the question: Compute the first five non-zero terms of the Taylor series about $a=4$ for ...
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3answers
244 views

Find an accurate value of $f(x)=\sqrt{4x^2+x}-2x$ for large values of x. Calculate $\lim_{x\to\infty}f(x)$

My works: $x^2$ can be very large if x is large, thus the function has lose-of-significance error and we need to reformulate it. $$ ...
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1answer
88 views

Use Taylor polynomials with remainder term to evaluate the following limits $\large\frac{e^x-x-1}{x^2}$

My work: Since $\large e^x=\sum\limits_{j=0}^\infty \frac{x^j}{j!}$, then $\large\frac{e^x-x-1}{x^2}=\sum\limits_{j=2}^\infty \frac{x^{j-2}}{j!}=\sum\limits_{d=0}^\infty \frac{x^{d}}{(d+2)!}$. (Let ...
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0answers
223 views

Taylor expansion: first derivative approximation with third order

About first derivative approximation with third order. Let $$f'(t)=\frac{(2t+h)\cdot{f(h)}-4t\cdot{f(0)}+(2t-h)\cdot{f(-h)}}{2h^2}+R.$$ Show that $$R=\frac{f'''(\xi)\cdot{(3t^2-h^2)}}{6}$$ and ...
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425 views

Taylor's Formula vs. Taylor's Inequality

In my calculus book, Essential Calculus, and in class we were using Taylor's formula to approximate the remainder in Taylor polynomials but I am having a bit of trouble understanding the intuition ...
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32 views

Optimization, descent direction, neccessary condition

I'm learning about nonlinear, unconstrained optimization. In my book it says that a descent direction $p_k$ must satisfy: $$p_k\nabla f(x_k)^T < 0$$ This seems to mean that $p_k$ must be obtuse to ...
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2answers
92 views

Series expansion of a function at infinity

I know it is possible to expanse an expandable fonction for a real, and for infinite by setting $x=\dfrac1y$ and then expanse for $0$. But my question is, how do we do if the evaluation of the new ...
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0answers
49 views

Taylor's Formula and 'z' values

I have attached Taylor's Formula, an exercise problem from a section on Taylor polynomials, and the solution to this exercise. I understand part a, expanding $f$ using Taylor polynomials is the ...
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1answer
44 views

Suppose $f \in C^{\infty}(\mathbb R)$ and $\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0.$ Show $f$ is analytic on $\mathbb{R}$.

Suppose that $f \in C^{\infty} (-\infty , \infty)$ and that $$\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0$$ for all $a\in \mathbb{R}$. Prove that $f$ is analytic on ...
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1answer
60 views

Second-order derivative wrt. vector

I have a scalar function $f(\mathbf{x})$, where its argument $\mathbf{x}$ is a vector. I am Taylor-expanding $f$, so I have to find $$ \mathbf{c}^2\frac{d^2}{d\mathbf{x^2}}f(\mathbf{x}) $$ where ...
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3answers
66 views

Proof that Polynomials Form a Basis

I'm not even sure this is a true statement, but can someone prove that the polynomials for a basis for continuous functions? This seems to be motivation for Taylor series, and several of the ...
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3answers
120 views

How to check the real analyticity of a function?

I recently learnt Taylor series in my class. I would like to know how is to possible to distinguish whether a function is real-analytic or not. First thing to check is if it is smooth. But how can I ...
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1answer
647 views

Taylor expansion of a vector field (notation question)

Is there an index-less notation (using gradiends, Jacobians, curls, hessians, anything) to describe a second-order term in the Taylor expansion of a vector field $\mathbf{f}(\mathbf{x}): \mathbb{R}^n ...
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2answers
57 views

Estimate Interval of Validity of $1-\frac{x^2}{2}$ for $\cos(x)$

I have been struggling with the following problem and was wondering if anyone could provide some insight or suggestions: Use $1-\frac{x^2}{2}$ as an approximation to $\cos(x)$, with an error not ...
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1answer
125 views

How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...
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1answer
68 views

Find “singular expansion” of a function

I have the function $(1-z)^{-z}$, analytic except on $\mathbb{R}_{\geq 1}$ Now in the text, it says the "singular expansion" at $z=1$ is $\displaystyle \frac{1}{1-z} + \log(1-z)+O((1-z)^{1/2})$ I'm ...
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1answer
26 views

Taylor polynomial aproximation - Interval of convergence

It is reaquired to find the Taylor polynomial of order $n$ of the cosine function around $x=0$. Then, it asks to find the biggest interval in which the sequence pf polynomial $p_n$ converges to $f(x) ...
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2answers
36 views

Does |Taylor Series of $f$ - $f$| Converge Monotonically to $0$?

Suppose that $T_n(x)$ be the sum of the first $n$ terms of the Taylor series of $f$ centered at $a$, and $\lim_{n\to \infty} T_n(b)=f(b)$. Is the difference $|T_n(b)-f(b)|$ decrease monotonically? ...
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1answer
90 views

Taylor theorem doubt(sin(x+h))

I was studying Taylor theorem when I came across this question in one of my math text books Obtain Taylor's series expansion of the function $\sin(\frac {\pi}{4}+h)$ in ascending powers of $h$. ...
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2answers
86 views

Taylor Series for $\log(x)$

Does anyone know a closed form expression for the Taylor series of the function $f(x) = \log(x)$ where $\log(x)$ denotes the natural logarithm function?
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2answers
207 views

Evaluating $\int_{0}^{\frac{\pi}{2}} \arctan( a \sin x) \ dx$ using the Taylor expansion of $\arctan (x)$

I was wondering if it's possible to show that for $a >0$, \begin{align}\int_{0}^{\pi/ 2} \arctan (a \sin x) dx &= 2 \sum_{k=0}^{\infty} \frac{\left(\frac{\,\sqrt{\vphantom{\Large A}\,1 + ...
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0answers
53 views

Computation of the remainder term on a Taylor expansion using contour integrals

I am not really used to the methods of complex analysis, I would like to know for basic monotonic functions like exp(x), log(x), sqrt(x), powers (x^n) and trigonometric functions defined on an real ...
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1answer
1k views

Find the Maclaurin series for $\cos(2x)$ using the series for $\sin(2x) $.

I know that $$\sin(2x)= 2x - \frac{8x^3}{3!} + \frac{32x^5}{5!} - \frac{128x^7}{7!} + \cdots $$ $$\sin(2x)= \sum_{n=0}^\infty (-1)^n {2^{2n+1}x^{2n+1} \over (2n+1)!}$$ But I don't see how I can use ...
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1answer
94 views

Taylor Polynomials, Why only Integer Powers?

So It seems that the definition of polynomial is that is is raised to an integer power, but why is this necessary? My question mainly arises from a proof of the solution to the Hydrogen atom in ...
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65 views

Radius of convergence for Taylor series?!

Given is: $f(x) = \frac{\sin x}{x} $ I need the Taylor series in $a = 0$, so: $$T(x,0) = \frac{1}{x} \sum_{n=0}^\infty ((-1)^n* \frac{x^{2n+1}}{(2n+1)!} ) = \sum_{n=0}^\infty (-1)^n * ...
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2answers
429 views

Sine taylor series

I'm pretty convinced that the Taylor Series (or better: Maclaurin Series): $$\sin{x} = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$ Is exactly equal the sine function at $x=0$ I'm also pretty sure ...
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1answer
86 views

exp(x) for imaginary numbers

Well, I know how to get the $e^x$ function polynomial expansion, but how do I know that this is also valid for imaginary numbers, like $i\pi$? I know that the ...
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1answer
34 views

Series — Coefficient Cn and Radius of Convergence

. I'm lost, and my textbook is failing me
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1answer
68 views

Lagrange Taylor remainder: can we choose $t^*$ continuously?

The Taylor theorem with Lagrange remainder tells us that for $f: \mathbb{R}^n \to \mathbb{R}$ twice differentiable (we can assume $C^2$ if we like), $$f(y) - f(x) = \left\langle \nabla f(x), y-x ...
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1answer
58 views

Weierstraß approximation on the real line

First of all: I am aware of the thread Weierstrass approximation does not hold on the entire Real Line. My question is just that if we have a function like $sin(x)$ that can be approximated by its ...
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1answer
268 views

Using partial fraction for $\cot \pi z$ to compute infinite sum

I want to compute the values $\sum_{n=1}^\infty \dfrac{1}{n^2}$ and $\sum_{n=1}^\infty \dfrac{1}{n^4}$ and $\sum_{n=1}^\infty \dfrac{1}{n^6}$ by comparison to the partial fraction development of $\cot ...
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0answers
136 views

Maclaurin series of $\sin(2\pi x)$

Find the Maclaurin Series for $$f(x) = \sin ( 2 \pi x )$$ using the definition of a Maclaurin series. If $f(x) = \sum_{n=1}^\infty c_{2n+1}x^{2n+1}$, give $c_{2n+1}$: ...
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1answer
150 views

Taylor series with Fibonacci coefficients

Let $\{a_n\}$ be the Fibonacci numbers given by $a_0=0,a_1=1,a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$. Prove that $f(z)=a_0+a_1z+a_2z^2+\ldots$ is a rational function, and determine which rational ...
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0answers
28 views

Taylor expansion with complex valued number

I like to do a Taylor expansion with a complex valued number, e.g.: $f(z) = \frac{1}{(1-\mathrm{i}z)} \qquad z \in \mathbb{C}$ Is there any restriction to expand this (let's say around $z=0$) ...
3
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1answer
216 views

Maclaurin series for $e^x +2e^{-x}$

I'm currently stuck on the question regarding the Maclaurin series for $e^x +2e^{-x}$ I've found that the power series representation for it is $$\sum_{n=0}^\infty \dfrac{x^n + 2(-x)^n}{n!}$$ ...
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2answers
54 views

Taylor's formula in several

What i'm puzzling about is this: I use the following form of Taylor's formula with integral remainder term: For a smooth function $\varphi$ it holds that $$ \varphi(x+h) = \varphi(x) + \int_0^1 ...
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0answers
165 views

Fractional Derivative of a Taylor Series?

I have a function defined only by it's taylor series: $f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!}x^k$ Obviously, integer derivatives can be defined as $\frac{d^n}{dx^n} f(x) = \sum_{k=0}^\infty ...
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1answer
103 views

Evaluate $I=\int_{\gamma (0,1)}\frac{zdz}{(z^2-4z+1)^2}$ using Taylor's Theorem

I've shown that $f(z)=\frac{z}{(z^2-4z+1)^2}$ is holomorphic apart from at points $\alpha=2-\sqrt3$ and $\beta=2+\sqrt3$ and that the talyor coefficient of $g(z)=\frac{z}{(z-\beta)^2}$ centred at ...
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1answer
25 views

Error estimation help

I'm supposed to find a Taylor polynomal of the $n^{\text{th}}$ degree, where $x = a$, and estimate the error for the given interval. The problem I'm given is: $$f(x) = \sqrt{x}, a = 4, n = 2, 4 \leq x ...
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2answers
83 views

Taylor/Maclaurin Series Exam Question.

Show that if x is small compared with unity, then $$f(x)=\frac{(1-x)^\frac{-2}{3}+(1-4x)^\frac{-1}{3}}{(1-3x)^\frac{-1}{3}+(1-4x)^\frac{-1}{4}}=1-\frac{7x^2}{36}.$$ I've expanded all the brackets ...
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3answers
62 views

Taylor/Maclaurin Series

Show that if x is small compared with unity, then $$f(x)=\frac{(1-x)^\frac{-2}{3}+(1-4x)^\frac{-1}{3}}{(1-3x)^\frac{-1}{3}+(1-4x)^\frac{-1}{4}}=1-\frac{7x^2}{36}$$ In my first attempt I expanded all ...
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1answer
74 views

Second partial derivation of vector function and taylor series

I have vector function: $$ f(x,y) = \begin{pmatrix} (R+r\cos(y))\cos(x) \\ (R+r\cos(y))\sin(x) \\ r\sin(y) \end{pmatrix} $$ I have done Jacobian of that function: $$ f'(x,y) = \begin{pmatrix} ...
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2answers
108 views

how to find the asymptotic expansion of the following sum:

I need to determine an asymptotic expansion when $q \rightarrow 1$ of the sum $$S(q)=\sum_{n=0}^{\infty} \frac{q^n}{ (q^n + 1)^2 }.$$ Numerical computations suggest that $S(q)\sim\frac{c}{|q-1|}$ ...
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2answers
163 views

How to know if a MacLaurin/Taylor Series expansion is good?

This question is motivated by this question. So, given $\frac{1}{e^x + 1}$, the 4th order MacLaurin series $1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4$, although correct in terms of the algebra manipulations, is ...
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1answer
304 views

Help find the MacLaurin series for $\frac{1}{e^x+1}$

What is the MacLaurin series up to $x^4$ for $\frac{1}{e^x+1}$? My Attempt: $$\begin{align} \frac{1}{e^x+1} &=(1+e^x)^{-1} \\ &\approx 1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4 \\ \end{align} $$ Since ...
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1answer
29 views

Maclaurin series and expressing as a ln(argument)

Found this question in my old homework notes that I did not do at the time! I always wondered how I do this... The first part is a explanation. It is kind of long. Sorry! Here is the actual ...
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0answers
81 views

Whats wrong in following attempt to write Hermite polynomials in terms of hypergeometric function?

Let's have Hermite polynomials: $$ e^{2tx - t^{2}} = \sum_{n = 0}^{\infty}H_{n}(x)\frac{t^{n}}{n!}. \qquad (1) $$ I need to write it in terms of confluent hypergeometric Kummer function for index $n = ...
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3answers
39 views

Maclaurin series for $ f(x)=a^x$

My friend is having trouble with these two questions on his homework. I want to help him out but I am not 100% sure how to do these. I took Calculus 3 a while back so its all old memory to me! ...