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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Binomial Expansion problem error

I tried solving this question but failed. a) Expand $(1+2x)^{1/4}$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each term as far as possible. b) By substituting ...
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1answer
117 views

Determine the Maclaurin series expansion of $\frac{\mathrm{exp}(z)}{(z+1)}$

Determine the Maclaurin series expansion of $\frac{\mathrm{exp}(z)}{(z+1)}$. This is the composition of the series expansion of the exponential function centered about $z = -1$. We can rectify the ...
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1answer
48 views

Estimate the degree of a Taylor Polynomial using its Error Term

In my 2nd year studying Maths at Uni and revising for a Numerical Analysis final exam. We're given 1 past paper but no solutions, and I can't answer this question: Use the error term of a Taylor ...
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1answer
317 views

$x^3-3x-3=0$, prove that $10^x<127$

$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$ I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
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1answer
173 views

Taylor series and tempered distributions

Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid? When we interpret ...
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1answer
185 views

Finding the error of the Taylor expansion of $\log(1 + x)$

The questions is as defined below. Let $f(x)= \log(1+x)$. Show that the Taylor remainder $R_{0,k}(x)$, defined by $$R_{a,k}(x)= f(a+x) - P_{a,k}(x) = f(a+h) -\sum_{j=0}^{k} \frac ...
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2answers
99 views

How does one get the Bernoulli numbers via the generating function?

Here is the definition: Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$ I've tried to naively expand $\frac{x}{e^x-1}$ around ...
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1answer
113 views

Itō's Lemma neglecting terms

In my project I am trying to give a Heuristic proof of Itō's lemma. I show $E[dW_t^2] = dt$ I take $g(x,t)$ to be a twice continuously differentiable function and $dt$ to be infinitesimally small. ...
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1answer
128 views

Integral remainder converges to 0

I want to show that $\displaystyle \log(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$ for $-1<x\leq1$ i want to show it with the integral remainder of the taylor series that gave me: ...
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2answers
334 views

taylor expansion of an integral $\int_0^1{e^{x^2}}$

I need to calculate $\int_0^1{e^{x^2}\:dx}$ with taylor expasin in accurancy of less than 0.001. The taylor expansion around $x_0=0$ is $e^{x^2}=1+x^2+\frac{x^4}{3!}+...$. I need to calculate when the ...
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286 views

Prove that d/dx (sin x) = cos x, using Taylor series

Show by differentiation of the series for sin x that $$\frac{d}{dx} (\sin x) = \cos x$$ (Using Taylor series.) If you can given an indication or solved answer to my question would be great. Thanks ...
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82 views

Determine the series for cos x^2

Use the series for Cos x (Taylor Series) If you could give me help or the solution to the problem, that would be great! Thanks
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1answer
48 views

Points around which one expands and the radiuses of convergence

I'm trying to make sense of the following passage: Let $f(x)=\frac{1}{x+1}$ and $R_0$ the radius of convergence of the Taylor series of $f$ around $x_0=0$, analogously: $R_1$ — around ...
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2answers
82 views

Multiplication of two Taylor expansions

I'm trying to calculate a Taylor expansion which is : $\cos(x). exp(x)$ in the neighborhood of 0 in order 3 this is the result I got : $$\cos(x). exp(x) = ...
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3answers
555 views

Taylor Series for $e^x$ where $x = 1$, estimating the Error

I'm trying to calculate $e$ to a certain number of digits. The Maclaurin Series expansion of $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. When $x = 1$ we can approximate the value of $e$ by ...
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549 views

In Taylor series, what's the significance of choosing the point of expansion $x=a$?

So I read about the Taylor series and it said you can choose to expand the series around a given point ($x=a$). Does it matter which point you choose in calculating the value of the series? For ...
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2answers
172 views

A question on the convergence of a Taylor series of some prominent function

The function $f:\mathbb{R}\to\mathbb{R}$ defined by $$ f(x)=\begin{cases}e^{-\frac{1}{x^2}} &if &x\neq 0\\ 0 & else \end{cases} $$ is a prominent example of a function whose Taylor series ...
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1answer
485 views

Taylor series with function composition

Pretty simple, but I want to take the first order taylor series expansion of the following: $f(g(x,y+Δy))$ Would the following be correct? $f(g(x,y+Δy)) = f(g(x,y) + \frac{\partial}{\partial ...
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1answer
80 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
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154 views

Taylor series expansion example

I was reading an article and there was a snippet with a taylor series expansion as shown below: My question is, should (11) read as $F(xA+h)+(xΔA+Δh)\frac{\partial}{\partial x}F(xA+h)$ instead of ...
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178 views

An issue with approximations of a recurrence sequence

By trying to give an approximation to a given recurrence sequence I encountered a problem. To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
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1answer
393 views

Taylor Series Expansion with e and sin

Show that when $z\neq0$, (a) $$\frac{e^z}{z^2}=\frac{1}{z^2}+\frac{1}{z}+\frac{1}{2!}+\frac{z}{3!}+\frac{z^2}{4!}+...$$ (b) ...
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2answers
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What's the Maclaurin series for $\arcsin(x)$?

I solved the problem by using a known series: $\frac{1}{\sqrt{1-x^2}}$, but the solution I got is wrong. Also, I'm not sure what to do with the constant of integration $C$. Where is my mistake? $$ ...
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1answer
77 views

Maclauren Series and taylor polynomials

Question: Suppose that the function $k(x)$ has a maclauren series that converges $\left(-\frac{1}{2} , \frac{1}{2}\right]$ and you are told that $|k^{(n)}(x)| \leq 10$ at all $|x| \leq ...
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1k 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 ...
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1answer
133 views

Taylor polynomial approximation

How do you determine if adding more terms to the Taylor polynomial will improve its approximation of $f(p)$ or in other words, how do you determine if a Taylor series converges for a particular value ...
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1answer
357 views

Using the series of $\tan^{-1}(x)$ for calculating $\pi$

The power series expansion of $\tan^{-1}(x)$ is $$\tan^{-1}(x)=x-\frac 13 x^3+\frac 15 x^5-\frac 17 x^7+ \cdots .$$ Use the above series to determine a series for calculating $\pi$.
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375 views

Analytic functions of a real variable which do not extend to an open complex neighborhood

Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
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3answers
301 views

Find Maclaurin series for$f(x) = \frac{2x}{1-5x^3}$

I'm trying to find the Maclaurin series for $f(x) = \frac{2x}{1-5x^3}$, but my solution is different from what I know it supposed to be, which is $2x+10x^4+50x^7+250x^{10}+...$ This is my attempt: ...
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1answer
128 views

Why do power series converge to a function symmetrically?

Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$? The selected answer to the above question says that for a a power series, the interval of convergence for the ...
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1answer
154 views

Derivative of a little-o remainder

If we have a $\phi: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$, $\phi = \phi(t, \mathbf{q},\alpha)$ one-parameter group of infinitesimal transformation which is $\mathcal{C}^2$ ...
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1answer
66 views

What is the exponential series representation of $x^x$?

I want to express $x^x$ in the form of an infinite series involving $qe^{sx}$ where $q$ is the $s$th coefficient of the series and $s$ is the power on $e^x$. Beyond just an answer I would like to know ...
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2k views

How to expand this taylor series and find radius of convergence

f(x)= √(1-x) at x=0 How do you find the taylor series and radius of convergence?
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1answer
266 views

How do you find Taylor series and radius of convergence for $\sqrt{x}$?

How do you find interval and radius of convergence of $f(x)=\sqrt{x}$ at $x=1$
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36 views

Demonstrate that $\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6}\ge 0$

How do I prove the inequality? $$\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6} \geq 0$$ I can see that $e^e = \sum_{k=0}^{\infty} \frac{e^k}{k!} = 1 + e + \frac{e^2}{2} + \frac{e^3}{6}+\dots ...
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51 views

Show that $|x-\ln(1+x)| \leq cx^2$

Use Taylor's Theorem to show that there is some fixed constant $c>0$ such that $$|x-\ln(1+x)| \leq cx^2$$ for all $|x|<\frac{2}{3}$. My attempt: Let $f(x)=\ln(1+x)$. Then by calculating the ...
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2answers
95 views

Taylor series of $f(x^2)$

If you know the taylor series for $f(x)$ can you find the taylor series for $f(x^2)$ by letting $x = x^2$? The taylor series in question is $\cos(x^2)$ I know the taylor series for $\cos(x)$ is ...
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1answer
93 views

Question about taylor series.

let $$f(x) = \begin{cases} \frac{\cos x -1}{x^2} & \text{for } x \neq 0 \\ \\ \\ -\frac{1}{2} & \text{for } x = 0 \end{cases} $$ The Taylor series for this is $$\dfrac{1 - ...
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3answers
121 views

$k$th term of Taylor series of function $f(x) = 3x^3-2x+4$

I'm having trouble solving this question. I have all the values right but can't figure out a way to find the $k$th term. My Taylor series values up to that point are ...
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1answer
188 views

expression for Remainder in Taylor theorem for complex variables

I don't understand how the following summation vanishes with $j=k+1$ in wikipedia article .$$R_k(z) = \sum_{j=k+1}^\infty \frac{(z-c)^j}{2\pi i} \int_\gamma \frac{f(w)}{(w-c)^{j+1}}dw = ...
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Finding the Maclaurin series representation

$$f(x)=\frac { x }{ { (2-x) }^{ 2 } }$$ I tried finding the first derivative, the second derivative, and so on, but it just keeps getting more complicated, so I suspect I have to use binomial series. ...
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120 views

Coefficient of Taylor Series of $\sqrt{1+x}$

The coefficient of $x^3$ in the Taylor series of the function $f(x) = \sqrt{1+x}$ about the point $a = 0$ is $$1\over 16$$ Can someone show me how to get this value?
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132 views

The coefficient of the Taylor series is given by?

what is considered the coefficient in a taylor series? and how would i solve this The coefficient of $(x-1)^3$ in the Taylor series of $ f(x) = ln x$ about $a=1$ is given by ?
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Taylor Series of $\sin 2x$ finding $f^{(n)} (a)$ where $a = 0$

ok so i get; f (x) = sin 2x f ' = 2cos 2x f '' = -4sin 2x f ''' = -8cos 2x f '''' = 16sin 2x f ''''' = 32cos 2x f (0) = 0 f '(0) = 2 f ''(0) = 0 f '''(0) = -8 f ''''(0) = 0 f '''''(0) = 32 ...
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1answer
131 views

Why do Maclaurin series approximate a function for negative domain values?

A common analogy used as an intuitive explanation for a Maclaurin series is that of a car. If you know the position, velocity, acceleration, jerk etc. of a car at time zero, you are able to predict ...
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90 views

$\frac{1}{1-x}$ series expansion

How do I know that the expression: $$\frac{1}{1-x}$$ Is equal to the infinite sum: $$-\left(\frac{1}{x}\right)-\left(\frac{1}{x}\right)^2-\left(\frac{1}{x}\right)^3-\left(\frac{1}{x}\right)^4+...$$ ...
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Basic Taylor expansions question

How can I use : $$f(x \pm h) = f(x) \pm hf^\prime(x) + \frac{h^2}{2} f^{\prime\prime}(x) + O(h^3)$$ to Prove: $$f^\prime(x)= \frac{f(x+h)-f(x-h)}{2h} + O(h^2)$$ Thanks in advance
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1answer
2k views

Difference between the Laurent and Taylor Series.

I have never taken complex analysis, but I am preparing for a GRE for this week end and I am trying to learn a bit about Laurent Series. So far what I get is that the Laurent Series are of form ...
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1answer
104 views

Don't understand how to start with this assignment question

I'm working on the last problem in an assignment, and need some guidance on what to actually start by doing. The question is asking me to use taylor expansion to determine the leading error term ...
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41 views

Inequality of Partial Taylor Series

For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k=0}^{N} \frac{x^k}{k!} ...