Questions tagged [taylor-expansion]

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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1answer
239 views

Problem with Picard Iteration

I have $ \frac{dy}{dx} = y^2, y(0) = y_0 $ I have solved this as $y = \frac{y_0}{1 - x y_0}$ Which has the Taylor expansion $ y_0+y_0^2 x+y_0^3 x^2+y_0^4 x^3+y_0^5 x^4+ ...$ However, when I ...
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4answers
188 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 expansions ...
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1answer
131 views

How could we rewrite this integral as the other one i provided and how to solve it? [duplicate]

Possible Duplicate: Value of $\sum x^n$ How do you rewrite this: $$ \int_{0}^{\infty} \cfrac{1}{1+x^4} $$ to: this$$\int_{0}^1 \sum_{n=0}^\infty (-1)^n x^{4n} + \int_{1}^\infty x^{-4} \sum_{n=0}^...
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2answers
759 views

The Taylor series of $\int_0^x \operatorname{sinc}(t) dt$

I tried to find what is the Taylor series of the function $$\int_0^x \frac{\sin(t)}{t}dt .$$ Any suggestions?
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4answers
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How many smooth functions are non-analytic?

We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather ...
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2answers
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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: $...
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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 ...
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1answer
150 views

Taylor polynomial for $f(x)=x^{1/7}$ about $a=1$

Here is the problem: (a) Determine the Taylor polynomial $T_2(x)$ of degree $2$ for the function $f(x)=x^{1/7}$ centered at $a=1$. (b) Suppose we were to use the approximation $f(x) \approx ...
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3answers
235 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 \limits_{n=0}^{\infty}\frac{3^n}{...
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1answer
162 views

Product of Taylor polynomials

I'm trying to prove the following proposition: Let $U\in R^n$ be open, and $f,g\colon U\to R$ be $C^k$ functions, then the Taylor polynomial of $fg$ is computed as $P_{f,a}^k(a+\vec{h})\cdot P_{g,a}...
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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
2k 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 $$\sec{z}=1+\sum_{k=1}^{\infty}{\frac{E_{2k}}{(2k)!}\,z^{2k}}$$ for ...
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0answers
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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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4answers
170 views

Calculating a Taylor Polynomial of a mystery function

I need to calculate a taylor polynomial for a function $f:\mathbb{R} \to \mathbb{R}$ where we know the following $$f\text{ }''(x)+f(x)=e^{-x} \text{ } \forall x$$ $$f(0)=0$$ $$f\text{ }'(0)...
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1answer
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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 $...
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2answers
162 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 geometric ...
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1answer
374 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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2answers
175 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 ...
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1answer
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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 $$\...
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2answers
194 views

Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$

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 $s=1$? ...
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3answers
513 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 \end{...
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2answers
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Using Taylor series expansion as a bound

I have a function $f(x)$ that has convergent Taylor series expansion around $x=0$ in the following form: $$f(0)-xg_1(0)+\frac{x^2}{2!}g_2(0)-\frac{x^3}{3!}g_3(0)+\frac{x^4}{4!}g_4(0)-\frac{x^5}{5!}...
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3answers
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Taylor series for different points… how do they look?

I can't understand what it means to do the Taylor series at the point $a$. The best way would be showing me how it looks for different $a$ on a graph. Do I find those graphs on the Internet?
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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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1answer
2k 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
283 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 ...
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1answer
560 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 $\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$. ...
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3answers
658 views

A property of roots of the truncated series for $\sin(x)$

Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$ In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$. $\begin{...
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1answer
667 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
2k 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) = c +...
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0answers
239 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 _n\...
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832 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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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 "...
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1answer
762 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 about ...
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1answer
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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 $δr^T$...
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117 views

Little question about finding a MacLaurin expansion for $f(x)=\frac{x^2}{1-x}$

First off all, I am sorry if my english is not perfect. I need help again for this exercise: Find Maclaurin series expansion for $f(x)=\frac{x^2}{1-x}$. That's what I did: $f(x)=\frac{x^2}{1-x}=\...
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1answer
191 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) ...
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1answer
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Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get decent starting points for some root finding algorithm to the roots of the scaled truncated taylor series of $\exp$. Here W ...
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3answers
280 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 ...
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4answers
373 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 ...
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1answer
578 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 ...
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3answers
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Is Fourier series an “inverse” of Taylor series?

I've understood Taylor series as being the representation of a "transcendental" function, using power functions with coefficents represented by appropriate derivatives. (Or maybe it is the MacLauren ...
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1answer
484 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} =\sum_{n=0}^\infty(-1)^nx^{...
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1answer
162 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 $$...
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1answer
3k views

Truncation error using Taylor series

How can we use Taylor series to derive the truncation error of the approximation $$f^\prime(x)\approx\frac{f(x+h)-f(x-h)}{2h}$$
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3answers
656 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 ...
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6answers
20k views

Don't understand why this binomial expansion is not valid for x > 1

today I'm studying binomial expansion and I'm a little confused about when certain expressions are valid. E.g. take this solution from my textbook: I understand that $(1-x)^{-1}$ has an infinite ...
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2answers
175 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
1k 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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3answers
168 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 $x < 1$.