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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4
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Taylor polynomial of $\int_{0}^{x}\sin(t^2)dt$

I just learned about Taylor polynomials, and I am trying to estimate $\int_{0}^{1/2}\sin(x^2)dx$ using the 3rd degree Taylor polynomial of $F(x)=\int_{0}^{x}\sin(t^2)dt$ at $0$. I get the following: ...
11
votes
4answers
1k views

On what interval does a Taylor series approximate (or equal?) its function?

Suppose I have a function f that is infinitely differentiable on some interval I. When I construct a Taylor series P for it, using some point a in I, does f(x) = P(x) for all x in I? I'm confused as ...
2
votes
5answers
1k views

Maclaurin polynomial of $\ln(\cos(x))$

I want to write down $\ln(\cos(x))$ Maclaurin polynomial of degree 6. I'm having trouble understanding what I need to do, let alone explain why it's true rigorously. The known expansions of ...
-1
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2answers
9k views

Taylor series of $\sqrt{1 + x^2}$

I want to know what is the Taylor series of $\sqrt{1+x^2}$ while $x \to 0$.
41
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2answers
1k views

Is there a function with the property $f(n)=f^{(n)}(0)$?

Is there a not identically zero, real-analytic function $f:\mathbb{R}\rightarrow\mathbb{R}$, which satisfies $$f(n)=f^{(n)}(0),\quad n\in\mathbb{N} \text{ or } \mathbb N^+?$$ What I got so far: Set ...
1
vote
2answers
173 views

What function does $\sum \limits_{n=1}^{\infty}\frac{1}{n3^n}$ represent, evaluated at some number $x$?

I need to know what the function $$\sum \limits_{n=1}^{\infty}\frac{1}{n3^n}$$ represents evaluated at a particular point. For example if the series given was $$\sum ...
4
votes
1answer
794 views

Why Does Substitution In Taylor Series Work? [closed]

The examples given here for example, show that once you know the form of a taylor polynomial as a function of $x$, you can replace the $x$ with another function. It works when you work out the ...
4
votes
3answers
565 views

Taylor Series of Ratio of Bessel Functions

In attempting to solve a recursion relation I have used a generating function method. This resulted in a differential equation to which I have the solution, and now I need to calculate the Taylor ...
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2answers
499 views

Clever methods of computing Taylor series expansion of $\sec{z}$ about $z=0$. [duplicate]

Possible Duplicate: Showing that $\sec z = \frac1{\cos z} = 1+ \sum\limits_{k=1}^{\infty} \frac{E_{2k}}{(2k)!}z^{2k}$ Show that ...
1
vote
0answers
36 views

Approximation of a function with certain restrictions at problematic points

I can't compute a Taylor series of a function like $f(x)=\sqrt{x}$ to some order around $x_0=0$, because the derivative at that point doesn't exist. If I consider the taylor series $Tf$ at any ...
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vote
1answer
561 views

Taylor series of $z/\sin(z)$ at $z=0$ by utilization of $\frac{1}{\sin z } = \cot z + \tan \frac{z}{2}$

In the book it is written: By using$$\frac{1}{\sin z } = \cot z + \tan\frac{z}{2}$$ one can easily compute the Taylor series (of the holomorphical extension) of $\displaystyle\frac{z}{\sin z}$ at ...
2
votes
2answers
151 views

Two Taylor expansions of $\frac{1}{1+\sqrt{2-z}}$ about $z=0$

How do you start expanding this function $$f(z)= \frac{1}{1+\sqrt{2-z}}$$ into two Taylor expansions about $z=0$? The best I came up is to let $u=\sqrt{2-z}$ and then expand $f(z)$ as a ...
0
votes
2answers
150 views

Domain of convergence of $f^{-1}: \mathbb R ^N \mapsto \mathbb R^N$ taylor series

In another question, I ask about the topology of the singular manifold of the Jacobian. What i want to ask in here is about the radius of convergence of a Taylor series expansion of the inverse ...
1
vote
1answer
227 views

Taylor series expansion of $\sec(x +y^2)$

We have $f(x,y) = \sec(x+y^2)$ I want to find the first two non-zero terms of $f$ at $(0,0)$ starting by Taking the first few terms of $\cos x$ centered at zero, $1 - \frac{x^2}{2!} $ Using this ...
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vote
0answers
231 views

Convergence of the Taylor series for the sine function

I would like to know if the Taylor series for the sine function, $$\sin{x} = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots,$$ is convergent if the argument of the function, $x$, is expressed in ...
6
votes
1answer
1k views

Using the Taylor expansion for ${(1+x)}^{-1/2}$, evaluate $\sum_{n=0}^\infty \binom{2n}{n} a^n$

Using the Taylor expansion for $${(1+x)}^{-1/2}$$ we have $${(1+x)}^{-1/2}= \sum_{n=0}^\infty \binom{-1/2}{n} (x^n)$$ for $|x|<1$. But if $|a| <1$, how can we use the above fact to find ...
3
votes
1answer
117 views

Series around $s=1$ for an integral

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
5
votes
3answers
353 views

A deceiving Taylor series

When we try to expand $$ \begin{align} f:&\mathbb R \to \mathbb R\\ &x \mapsto \begin{cases} \mathrm e^{-\large\frac 1{x^2}} &\Leftarrow x\neq 0\\ 0 &\Leftarrow x=0 ...
2
votes
2answers
713 views

Help finding the absolute error with $n$th degree Taylor polynomials

I am trying to estimate the absolute error in approximating $\ln 1.09$ with the $3$rd-order Taylor polynomial centered at $0$. It's been a while since I've taken the Calculus and I'm afraid I need ...
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votes
1answer
224 views

What if $f^{(n)}(a)=0$ for all $n\geq 0$?

This morning I was trying to imagine what a function would look like if all it's derivatives were zero at a point $a$ (assuming it is $C^\infty$). My first thought was that it should be identically ...
1
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1answer
595 views

Approximate the integral $\int_0^1 \sin(x^2) dx$

I'd like to ask if someone can please give me a little push with this assignment: Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations ...
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1answer
237 views

Maclaurin series of $\frac{1}{1+x^2}$

I'm stumped here. I''m supposed to find the Maclaurin series of $\frac1{1+x^2}$, but I'm not sure what to do. I know the general idea: find $\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$. ...
0
votes
1answer
208 views

Really basic question about the Taylor expansion of a CDF

I am sorry for such a basic question... but I want to try to do a Taylor expansion on my function, which is a CDF defined over 0-1. However, when I expand around 0, which is what I read is typical, ...
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1answer
551 views

Taylor expansion and big O

The expressions $e^h, (1-h^4)^{-1}, \cos(h), 1+\sin(h^3)$ all have he same limits as $h\to 0$. Express each in the following form with the best integer values of $\alpha$ and $\beta$. $$f(h) = ...
0
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0answers
173 views

Expanding an expression using Taylor's series

We've been attempting to expand an expression with Taylor's Theorem but can't quite make the math work out. $$ \frac{f\left(x_n\right)}{f'\left(x_n\right)}= \frac{1}{m}\frac{f^{(m)}\left(\xi ...
5
votes
3answers
304 views

Taylor series $\ln(\tan(x))-\ln(x)$ for point $0$

Well. I want to find the Taylor series for the function: $$f(x) = \ln(\tan(x))-\ln(x),$$ order $5$ for point $c=0$. Maple's result is: $$\ln(\tan(x))-\ln(x) = \frac 13x^2+\frac 7{90}x^4+O(x^6).$$ ...
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0answers
1k views

Taylor series and little-oh notation

Consider the series $e^{\tan(x)} = 1 + x + \dfrac{x^{2}}{2!} + \dfrac{3x^{3}}{3!} + \dfrac{9x^{4}}{4!} + \ldots $ Retaining three terms in the series, estimate the remaining series using ...
0
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1answer
335 views

Need help understanding Hessian matrix for direction estimation

Additional context: $H = |δ^2f / δx_iδx_j|$ is the Hessian matrix. $(3)$ From my previous question: What are the functionality of δ symbol and $δr^T$?, I got a few questions: I have read more ...
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1answer
60 views

What are the functionality of δ symbol and $δr^T$?

I got two questions here: Does anybody know what is the functionality of the small delta letter δ here? Is it simply the same as the rate of change just like the big delta letter Δ? And for the ...
4
votes
1answer
153 views

Please help me to find Taylor expansion (or approximation) for $f(x)=\frac{1}{x^2(x-1)}$ around $a=2$

First, sorry if my translations is bad. I need help for this exercise, more precisely , I need to know if my result which I've found is good. The exercise: Find Taylor expansion (or approximation) ...
5
votes
2answers
426 views

Computing the odd terms of the Taylor series of $\frac{z}{e^z-1}$

I know that the terms are $0$ for odd $n > 1$, but I haven't had any luck proving this. Computing them directly verifies this for small $n$; the function is also analytic, so I've tried taking the ...
10
votes
3answers
265 views

Quick way to expand $\cos^{-1}(\cos^2 x)$ up to $O(x^2)$

For a part of a question, I need to expand $\cos^{-1}(\cos^2 x)$ up to $O(x^2)$ about $x=0$. It took me quite a while to get an incorrect answer. What are some quick and efficient offline (i.e, no ...
2
votes
4answers
212 views

Need help in Taylor series expansion

In this question, I have to write Taylor's series expansion of the function $f(x) = ln(x+n)$ about x = 0, where n ≠ 0 is a known constant. I have done the following: But my professor handed me back ...
3
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1answer
1k views

Truncation error using Taylor series

How can we use Taylor series to derive the truncation error of the approximation $$f^\prime(x)\approx\frac{f(x+h)-f(x-h)}{2h}$$
3
votes
1answer
295 views

Laurent series for an infinite sum of functions

I have an infinite sum of analytic functions that is guaranteed to converge for every $x$, except for $x=0$: \begin{equation} g(x) = \sum_{n=1}^\infty f_n (x) \end{equation} I want to expand the ...
19
votes
1answer
263 views

Are there always singularities at the edge of a disk of convergence?

Take a function that is analytic at 0 and consider its Maclaurin Series. Here are some examples I'll refer to: $$\frac{1}{1-x} =\sum_{n=0}^\infty x^n$$ $$\frac{1}{1+x^2} ...
2
votes
1answer
128 views

A question about the product of two series

Given two power series, $$f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$$ and $$g(x)=\sum_{n=0}^{\infty}b_{n}x^{n}.$$ It is easy to form their product $$f(x)g(x)=\sum_{n=0}^{\infty}c_{n}x^{n}$$ where ...
14
votes
3answers
365 views

Under what conditions is integrating over a series expansion valid for an improper integral?

On stackoverflow, a question was asked about getting Mathematica to evaluate the integral, $$\int^\infty_0 \frac{e^{-x}}{\sin x} \, \mathrm{d}x$$ which we know is divergent. In one of the answers, ...
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vote
2answers
130 views

How does this Taylor Polynomial work?

The Taylor Polynomial is defined as following: $$P_n(x) = 1 + \dfrac{1}{2}x - \dfrac{1}{8}x^2 + \cdots + (-1)^n \dfrac{1.3.5 \cdots (2n - 3)}{2.4.6 \cdots 2n}x^n$$ If $n = 4$, then the last term in ...
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3answers
165 views

How can I write $\frac{1}{(a+x)}$ as an exponential function $y = Ce^{-kx}$?

How can I write $\frac{1}{a+x}$, $a$ a non-zero positive constant, in exponential terms in the form of $y = Ce^{-kx}$? I've tried to use to Taylor series but they only seem to work for $x < 1$.
2
votes
0answers
193 views

4 vector Taylor expansion, sign confusion

I've been presented with a function expansion which I'm told is correct but I can't figure out where the sign in the second term might be coming from. $$ e^{i\alpha(x_\mu + \epsilon \, n_\mu)} = ...
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1answer
188 views

Stuck on Taylor expansion question

I was given this question by a friend: Let $f(x)$ be 2 times differentiable in $R$ and $x_{0}$ a local extremum. Show that there are $a, b, c \in R$ such that a function $g(x)=f(x+a)+b$ ...
2
votes
1answer
739 views

Coefficients for Taylor series of given rational function

Looking at an earlier post Finding the power series of a rational function, I am trying to get a closed formula for the n'th coefficient in the Taylor series of the rational function (1-x)/(1-2x-x^3). ...
2
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1answer
1k views

Proving Lagrange Form of Remainder for Taylor Polynomial

So I got to the infamous "the proof is left to you as an exercise" of the book when I tried to look up how to get the Lagrange form of the remainder for a Taylor polynomial. Is this right? Given ...
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1answer
227 views

Taylor expansion in time of the time component of a stress energy tensor

Perform a taylor expansion in 3 dimensions in time on the time compontent of of $T^{\alpha \beta}(t - r + n^{i} y_{i})$ given that $r$ is a contstant and $n^{i} y_{i}$ is the scalar product of a ...
3
votes
2answers
485 views

Taylor polynomial with Lagrange remainder

In my course there's a paragraph: Taylor polynomial with Lagrange remainder, The paragraph starts with a theorem (I left out the constraints): $ ( \exists \theta \in ]0,1[)(f(a +h) = T_{f,a,n}(a + h) ...
0
votes
1answer
142 views

Taylor Series. Reusing an approximation of a function

I have this function, $e^{-x}$ bounded between 0 and 1500 and I have an approximation by Taylor Series of the same function bounded between 0 and 0.5. I would like to express my function $e^{-x}$ ...
6
votes
1answer
125 views

Why is the remainder function $R_{n}(x)$ decreasing?

When solving questions like these: Let $f(x)$ be a real function. Find $f(0.1)$ using its Taylor expansion such that the error is less than $10^{-3}$. Find the lowest degree of Taylor ...
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1answer
316 views

How to express the whole part $\lfloor x \rfloor$ as analytical function or Taylor/Fourier series?

And how to express $\{ x \} = x - \lfloor x \rfloor$ as function of $sin(x)$ and $sign(x)$?
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1answer
486 views

Basic Taylor expansion question

I seem to have a misunderstanding of how to work with a Taylor series. Suppose I want to write $f(x)=x e^x$'s Taylor expansion of $n$ degree around $0$. I see two ways: 1) Find the $n$th ...