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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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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question about taylor series

Can someone explain why 1 and 2 use different Taylor series? Why i cant use $1/(1+r)$ = $\sum_{n=0}^{inf}(-1)^n r^n$ on 2,vice versa?
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What is the characteristic function used for?

Im totally new to statistics , but what is the characteristic function for ? I do not get that. I was reading about the bell curve and the Central Limit Theorem , but I did not get what the ...
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4answers
153 views

Infinite series expansion of $e^{-x}\cos(x)$

Establish an infinite series expansion for the function $y=e^{-x}\cos(x)$ from just the known series expansions of $e^x$ and $\cos(x)$. Include terms up to the sixth power. I know that the ...
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Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
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1answer
132 views

Laurent series, radii of convergence.

I'm working on the following exercise: Prove that a Laurent series \begin{align*} \sum_{n = -\infty}^\infty a_n(z-z_0)^n = \sum_{n = 0}^\infty a_n(z-z_0)^n + \sum_{n = 1}^\infty ...
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Counterexample: For real functions existence of all higher order derivatives doesn't imply analycity.

In the lecture we had an example for a function $f: \mathbb R \to \mathbb R$, which is not analytic. We defined, that a function is said to be analytic at some point $x_0$ if a Taylor series expansion ...
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f is a smooth function, and $M_n$ is the sup of $f^{(n)}$. Show if $\lim_{n \to \infty} \frac{M_n}{n!}R^n < \infty$, then f(x) is the taylor series.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function (i.e. assume that the n-th derivative $f^{(n)}$ is defined on all of $\mathbb{R}$). Let $R$ denote the radius of convergence of the Taylor ...
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Find the order of the following expression as x->0

Could someone help me find the order of the following expression without using the quotient rule? $\frac{1-\cos(x)}{1+\cos(x)}$ I expanded the denominator and the numerator but not sure how I get to ...
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2answers
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Third order term in Taylor Series

What is the third order term in the Taylor Series Expansion? I know it will just be third order partial derivatives but I want to know how is it expressed in a compact Matrix notation. For instance ...
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128 views

Find $a$ such that $\lim_{x\to 0} \frac{1-\cos(\sqrt{ax})}{x^2}=3$.

Find $a$ such that $$\lim_{x\to 0} \frac{1-\cos(\sqrt{ax})}{x^2}=3.$$ Can we solve it with l'Hospital's Rule or do we need to use Taylor series? I have tried using L'Hospital's Rule and i keep ...
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1answer
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Proving that 2 functions are equal/not equal

Prove the equality of $f_1$ and $f_2$ given the following conditions: Problem 1 $f_1(x)$ and $f_2(x)$ are functions of finitely summed sine and cosine functions (e.g. $3\cos2x+\sin5x$), any ...
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1answer
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Maclaurin series of $\ln(2+x^2)$

Find the Maclaurin series of $\ln(2+x^2)$. I know that $\displaystyle\ln(1+x) = \sum_{n=1}^\infty\frac {(-1)^{n-1}} {n} x^n $ So is $\displaystyle\ln(1+x^2) = \sum_{n=1}^\infty \frac ...
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How can I compute this limit? [duplicate]

I have to compute $$ \lim_{n\to\infty} \exp(-n)\left(1+n+\frac{n^2}{2}+\ldots+\frac{n^n}{n!} \right)$$ I think the value is 1, but i don't know how to proof this. Do I have to estimate the remainder ...
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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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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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1answer
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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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3answers
278 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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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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101 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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how is the gradient derived here?

I'm taking an online machine learning class and in lecture 9 which covers gradient descent, I can't quite follow how he derives the direction vector of the descent (around the 57:15 mark). He's ...
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698 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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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
111 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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3answers
2k views

radius of convergence of $1/(1+x^2)$

Which is the radius of convergence of Taylor series of $f(x)=\frac{1}{1+x^2}$? I am unable to write down the analitycal expression of all its derivatives. Why is it finite even if $f(x)<\infty$ ...
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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
78 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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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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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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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
152 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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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
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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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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
113 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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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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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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109 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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71 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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1answer
103 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
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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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An exception to Taylor Series

According to Taylor Series, $$f(x) = \sum_{n=0}^\infty \dfrac{f^{(n)}(a)}{n!}*(x-a)^n $$ However, $\dfrac{1}{x}$,$\dfrac{1}{x^2}$, etc. are not applicable. I tried to do the following: $$ ...
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479 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
615 views

Taylor's theorem for vector valued functions

I'm reading about linear and nonlinear programming and on one page I have the following statment (I have highlighted the areas where I have problems and drawn questions for them in the bottom of it): ...
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102 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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127 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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Taylor expansions at $x=\infty$

How do you expand, say, $\frac{1}{1+x}$ at $x=\infty$? (or for those nit-pickers, as $x\rightarrow\infty$. I know it doesn't strictly make sense to say "at infinity", but I think it is standard to say ...
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1answer
298 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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Practical Use of Series Expansion at $x=\infty$

Asking WolframAlpha on certain functions, it happens that you get a series expansion at $\infty$. Thinking of the expansion as an approximation of the function in the vincinity of a point $a$, like in ...