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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How to show the existence of an entire function

I have been working on this problem for quite sometime. For part (i), I obtained the Taylor series for $4\sin(z) - \sin(4z)$. At $z = -\pi$, the Taylor series is: $4\sum_{n=0}^{n} \frac{(z + \pi)^{...
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48 views

Compact Form of the Taylor Series

Determine the Taylor Series $\frac{1}{\sqrt{1-x}}$ at $x=0$ I ended up with this: $1 + \frac{1}{2}x+\frac{3}{4}x^2\frac{1}{2!}+\frac{15}{8}x^3\frac{1}{3!}+\frac{105}{16}x^4\frac{1}{4!}$ I am ...
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70 views

Asymptotics and little-o notation

I always have issues dealing with asymptotic notation... I am trying to verify the following step: $$\left(1-\frac{t^2}{2n} + o(1/n)\right)^n \to e^{-t^2/2}.$$ To change this into $(1-t^2/(2n))^...
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taylor expansion of $\sinh(x)$

I would like to find taylor expansion of $sh(x)$ My thoughts indeed, note that : $\sinh(x)=\dfrac{e^{x}-e^{-x}}{2}$ then \begin{align} \sinh(x)&=\frac{e^x-e^{-x}}{2} \\ &=\frac{1}{2}\...
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To show that the limit of the sequence $\sum\limits_{k=1}^n \frac{n}{n^2+k^2}$ is $\frac{\pi}{4}$

Show that $$\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{n}{n^2+k^2} = \frac{\pi}{4}.$$ I am familiar with Taylor series and Fourier series of the standard functions. I tried to compare with those ...
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38 views

Taylor series for df

So I understand if I have f(x) under a taylor expansion I can write the terms up to order 2 terms as: f(x)= f(a) + f'(a)(x-a) + [f''(a)*(x-a)^2]/2! +... so I would imagine df(x)/dx = f'(a) + [f''(a)...
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$\ln(1+x)\underset{x\to 0}{=} \sum_{k=1}^{n}(-1)^{k+1}\frac{x^{k}}{k}+o(x^{n})$

I would like to see the setps behind that implication $$ \frac{1}{1+x}\underset{x\to 0}{=} \sum\limits_{k=0}^{n-1}(-1)^{k}x^{k}+o(x^{n-1}) \implies \ln(1+x)\underset{x\to 0}{=} \sum\limits_{k=1}^{n}(...
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48 views

Leading order Taylor Series Represention of the following function

I am given with this function $$f=\frac{1}{\sqrt{1+af_1(x)+bf_2(x)}},$$ where $$f_1=(1+x^2)^\nu,$$ and $$f_2=x^2(1+x^2)^{\nu-1},$$ where $\nu$ is a rational constant. I would want my $f$ to be of the ...
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23 views

Reducing terms in the series expansion of a function of two variables

I have a function $f(x, y)$. This function is such that \begin{align} f(0, y)=a\\ f(x, 0)=a, \end{align} where $a$ is a constant. From this, a particular mathematician concludes: Thus if we ...
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48 views

Taylor expansion for vectors

$$F(x,y)= (x_2-x_1^2) (x_2-2x_1^2)= 2 x_1^4+x_2^2-3x_1^2x_2$$ Where $x^*=[x_1 \ \ x_2]' = [0 \ \ 0]'$ I want to show Taylor expansion of the function for third degree. What I did is that; ...
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75 views

Expand in Taylor series $\frac{1}{1-\sin{x}}$

Expand in Taylor series $\frac{1}{1-\sin{x}}$ I have an idea that $\frac{1}{1-\sin{x}} = 1 + \sin {x} + \sin^2 {x} + \sin^3 {x} + \dots$ But I don't know what to do next. Every sine expands in ...
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1answer
32 views

approximating $(1-e^{-x})^2$ near $x=0$ with $x^2$ via Taylor expansion

I would like to show that $(1-e^{-x} )^2$ is approximated well near $x=0$ with $x^2$ via Taylor expansion but can't quite seem to complete the job. I know that by expanding the exponential into its ...
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25 views

How to find the degree- n term in the Maclaurin polynomial of $f(x)=\ln(1+x)$?

How to find the degree- n term in the Maclaurin polynomial of $f(x)=\ln(1+x)$? My Thoughts: The nth term is obviously: $$\frac{f^{(n)}(0)}{n!}x^n$$ But I am stuck here, how do I find the nth ...
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22 views

Find the value of $n$ such that the Maclaurin polynomial error is within a bound

Let $T_n(x)$ be the $n^{th}$ Maclaurin polynomial for $f(x) = e^x$. Use the error formula to determine a value of $n$ so that $\lvert T_n(2)−e^2\rvert < 10^{−4}.$ I haven't seen a problem like ...
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13 views

Finding the Interval of Convergence for the Sum/Difference of two Power Series

The question: Find the Taylor Series for $$f(x) = \frac{1}{x^2-3x-18}$$ at x = 1. Find the interval of convergence. My work: $$\frac{1}{x^2-3x-18} = \frac{1}{9}(\frac{1}{x-6}-\frac{1}{x+3})...
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75 views

Help with arithmetic on basic Taylor Series expansion

There are two of the steps below that I would seek assistance on the arithmetic. From the wikipedia article on Taylor series: https://en.wikipedia.org/wiki/Taylor_series The Maclaurin series for $(...
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1answer
186 views

Cauchy Product of two Taylor Series

I'm probably being a bit stupid here but I've been assigned this question and don't really know where to go with it. Compute the first 5 terms of the Cauchy product of the Taylor Series for $(1-x)^{2/...
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49 views

Series expansion at infinity

I am trying to find to generalize the limit that involves all rational functions such as $\sum_{n=0}^{l}\frac{{a}_{n}{x}^{n}}{{b}_{n}{x}^{n}}$. I believe the best way of generalizing all of them is ...
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54 views

How to prove $\frac{x}{e^x-1}=1-\frac{1}{2}x+\frac{1}{12}x^2+o(x^2),(x\to0)$ using Taylor's Formula?

$$\frac{x}{e^x-1}=1-\frac{1}{2}x+\frac{1}{12}x^2+o(x^2),(x\to0)$$ I have attempted to expand the multinomial $e^x-1$ by using Taylor's Formula, and I got this: $$\frac{x}{e^x-1}=\frac{x}{x+\frac{x^2}{...
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Maclaurin series of $e^{-x^2}$ Error

The task is to first estimate the second degree Maclaurin series of $e^{-x^2}$ and thus estimate the integral of the function from $0$ to $0.5$. This part is no problem. The following task is to ...
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12 views

Bounding the error for $e^{x+5y}$ taylor polynomial expansion

The exercise asks me to prove: $$|e^{x+5y}-P_1(x,y)|< \frac{3}{2}(x+5y)^2$$ when $x+5y<1$ I don't understand what's the exercise suggesting but I tried this: $e^{x+5y} - P_1(x,y)$ is just ...
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85 views

On the binomial series $(1+\frac{1}{8n})^{1/2}$, where $n$ is an even perfect number

Since $\sqrt{1+8n}=\sqrt{8n}\sqrt{1+\frac{1}{8n}}$, and $\frac{1}{8n}<1$ when $n>1$ is an integer, then we can express the real number $\sqrt{1+\frac{1}{8n}}$ by its binomial series. This series ...
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30 views

Taylor polynomial of degree 2 of $e^{x^2+x}$

I want to find the Taylor polynomial of degree 2 of $e^{x^2+x}$ and this is what the answer should be: $$e^{x^2+x} = e^{x^2}e^{x} = (1 + x^2 + O(x^4)) (1 + x + \cfrac{x^2}{2} + O(x^3)) = 1 + x + \...
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Taylor expansion of difference of functions

Is the taylor expansion of the difference of functions (more specifically the difference of the same function at different points) simply the difference of the taylor expansions? Since that may be ...
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Taylor series of a convolution

The derivation below is from Probability Theory: The Logic Of Science By E. T. Jaynes, chapter 7 "The Central Gaussian, Or Normal, Distribution", p.706 The Landon derivation. Text available online: ...
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Estimating error using Taylor Polynomial

I have searched and read quite a bit on this subject but I can't get this last bit straight. Reading the other answers did not help me unfortunately for me. Anyway the problem: Suppose I have the ...
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63 views

Is e^(-1/x) a flat function at x = 0?

Taylor series of e^(-1/x) at x = 0 shows that it is flat function on x= 0. But in every text on flat function I see the example of the function e^(-1/x^2) and not e^(-1/x). I am starting to think that ...
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How large need n to be to ensure that Taylor polynomial around x=0 gives a value of sin(pi) which has an error of less than 0.001?

I've found different methods to calculate $n$, but all include that I test it for several $n$. Is it possible to make a general formula that gives me the answer without having to test it, or do I need ...
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1answer
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expressing $p , p(p+1) , p(p+1)(p+2)$ as a series

I'm working on arithmetical analysis and more specifically on finite differences. I want to create a series consisting of the following terms : $$f(x_{0} + ph) = f_{0}+ p\Delta f_{0}+ \frac{p(p+1)}{2!...
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Taylor Series Expansion of $ f(x) = \sqrt{x} $ around $ a = 4 $

guys. Here's the exercise: find a series representation for the function $ f(x) = \sqrt{x} $ around $ a = 4 $ and find it's radius of convergence. My doubt is on the first part: I can't seem to find ...
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46 views

Prove that the series $ \sum_{n=0}^\infty \frac{(-1)^n \cdot x^{2n}}{(2n)!} $ represents $ \cos x $ for all values of $ x $

guys. The question is as stated in the title: prove that the series $ \sum_{n=0}^\infty \frac{(-1)^n \cdot x^{2n}}{(2n)!} $ represents $\cos x $ for all values of $ x $ My doubt is quite theoretical:...
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19 views

Write the indicated case of taylor's formula

I have this problem: "Write the indicated case of Taylor's formula for the given function. What is the Lagrange remainder in each case? $f(x) = \ln{x}$ $a = 1, n = 6$ " That's the information I ...
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26 views

Is there a way to separate this function?

let $f(\mathbf r_1,\mathbf r_2) = \frac{1}{|\mathbf r_1 - \mathbf r_2|^2 + a^2}$. Is there a method to represent this as (a series of) separated functions in the form?: $f = \sum \limits _i g_i(\...
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35 views

How to derive analog of power rule for other forms of the derivative?

Introduction We'll be dealing with multiple forms of calculus here. So we'll use $\operatorname{L_d}(f(x))$ to refer to the additive derivative, $\cfrac{df}{dx}$, $\operatorname{P_d}(f(x))$ to refer ...
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What is the trick to Taylor expand this function to 4th order?

The function is $ u(x,y)= -x-y-xyu^3$, and I want to Taylor expand $u(x,y)$ around (0,0) in powers of x and y to 4th order. To first order, I differentiated implicitly, and the expansion is: $$u(x,y)...
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remainder term error in maclaurin polynomial

Consider function f(x)=$\frac{1}{1-x}$, find the remainder term Rn(Z) of a function of x and n. I now know that $f^{(n)}(x)=\frac{n!}{(1-x)^{n+1}} $ and that $Rn(z)=\frac{f^{n+1}(z)}{(n+1)!}(x)^{n+1}$...
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Is the remainder of first-order Taylor expansion still continuously differentiable?

Let $f: {\mathbb R}^n \to {\mathbb R}^n$ be a continuously differentiable function. Then, we can rewrite its first-order Taylor expansion at $x \in {\mathbb R}^n$ for $h \in {\mathbb R}^n$ that \begin{...
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Expanding a Function into a Maclaurin's Series

"Given that $y^3=e^xcosx$, show that $3y^2\frac{dy}{dx}-y^3=-e^xsinx$. By further differentiation of this result, or otherwise, find the Maclaurin series for y, up to and including the term in $x^2$."...
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Limit is infinite or finite?

The given function is $${{Log\ z}\over {z-1}}=1- {1\over 2}(z-1)+ {1\over 3} (z-1)^2-{1\over 4}(z-1)^3+.... $$ Then it is said that the function tends to $+\infty$ as $z$ tends to $0$ . But ...
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How to use Taylor series to get $e^x\geq1+x$

I know that from $$e^x=\sum_{i=0}^\infty \left(\frac{x^i}{i!}\right)$$ we can get the inequality $e^x\ge1+x$. But how?
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How do I evaluate this without using taylor expansion :$\lim_{x \to \infty}x^2\log(\frac {x+1}{x})-x\ $?

How do I evaluate this without using Taylor expansion? $$\lim_{x \to \infty}x^2\log\left(\frac {x+1}{x}\right)-x$$ Note: I used Taylor expansion at $z=0$ and I have got $\frac{-1}{2}$ Thank ...
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41 views

Radius of convergence of a Taylor-series

I came across the following question. Let $c \in \mathbb{R}$ and let $f: \mathbb{R} \to \mathbb{R}$ be defined by: $$ f(x) = \frac{1+ c x^2}{1+ x^2}$$ Let $c \neq 1$. Determine for $a=0$ the ...
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Functions on a connected subspace

For one of my courses I came across the following question. Let $U$ be a connected supspace of $\mathbb{C}$ and let $f: U \to \mathbb{C}$ and $g: U \to \mathbb{C}$ be complex analytic (eg complex ...
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23 views

Can I have a proof of the Taylor Series of $f$?

My textbook states the following: Taylor Expansion: $f(x)=\frac{f(a)}{0!}+\frac{f^{\prime}(a)(x-a)}{1!}+\frac{f^{\prime\prime}(a)(x-a)^2}{2!}...$ I actually have used this a lot in my own personal ...
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83 views

How to prove the following Taylor expansion for twice differentiable functions

I want to prove the following: Let $f(x)$ be a twice differentiable function. Then, $$\begin{array}{l} \exists t \in \left[0,1 \right ] \; s.t., \\ f\left(y \right ) = f\left(x \right ) + \left(y-x \...
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24 views

Evaluate limit of function through its series expansion

The MacLaurin series expansion of the function $$f(x) = \frac{1}{1 - x}$$ is $$f(x) = 1 + x^2 + x^3 + x^4 + \ldots = \sum_{i = 0}^{+\infty} x^i, \ (x \neq 0)$$ so all the powers of $x$ taken with ...
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1answer
195 views

Mistake on a Major Maths Website

I think I have found two massive errors on Math.com but I throw my logic out here for a third party to verify as I've been doing Contour Integrals for about 10 hours straight now so I am very tired. ...
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1answer
64 views

Find $n$ such that $\tan 1$ and its Taylor series up to $n$ agree to 1000 decimal places

I know the Taylor series for $\tan x$ is, $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$ I am trying to find a value for $n$ such that $|\tan 1 -\...
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22 views

Differential equation with expansion

In this problem of physical origin (Newton's equation of motion of a charged particle in an electric field), can you help me with equation (1.7)? I get more or less where it comes from, but I could ...
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1answer
41 views

Surprising pattern of big primes in a derivative series

A pattern of large prime numbers appeared where I was not expecting one, and I'm fascinated to know why. I needed to calculate the Taylor Series for $$f(x)\equiv\ln\left(e^{(x+1)^{2}}+e^{(x-1)^{2}}\...