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.
8,567
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I do not know why there is a zero in this formula.
I am currently studying the chapter on Numerical Differentiation in my book. However, I have encountered a section where a zero appears in the expansion, and I'm unsure of the reason behind its ...
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Approximating norm using Taylor series
I have a point $A =(r\cos\theta \sin\phi, r\sin\theta \sin\phi, \cos\phi) $ and $B =(nd_x, 0 ,md_z)$. My aim is to compute the Euclidean norm $||A-B||$. However, I am interested in an approximation ...
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Taylor Expanding a higher power of a function
I have a (sufficiently smooth) function $f$ that is volume-preserving (Jacobian has determinant $1$) and invertible. Given two integers $\ell, k \in \{0, 1, \ldots, T\}$ with $\ell > k$ it seems ...
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Integrating $\int_0^1 x^xdx$ [duplicate]
How would you calculate the following integral?
$$\int_0^1 x^xdx$$
I write my answer below. Don't be shy to speak out if you find the mistakes (if there are any) or have a better solution.
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Taylor Expansion Subseries
If we Taylor expand an infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ around a point $a \in \mathbb{R}$ we of course get:
$$ f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(...
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Lagrange's Remainder for the Taylor formula of $\sin(\cos(t))$.
First I am asked to compute the following:
Show that for $0 < x < \frac{\pi}{2}$ we have:
$\sin(x) = x - \frac{x^3}{3!} + R(x)$ and $0 < R(x) < \frac{x^5}{5!}$
This part can be done simply ...
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Convergence of taylor series expansion in the evaluation point
The Taylor series expansion of $f(x)=e^x$ in $a=1$ is:
$T(x,a=1)=\sum_{n=0}^{\infty}\frac{e\left(x-1\right)^{\ n}}{n!}$
The interval of convergence using the absolute ratio test is $x\in(-\infty,\...
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How can I discretly derive $u_{n}=\sin(np)$ where $p>0$?
How can I discretly derive $u_{n}=\sin(np)$ where $p>0$? I would show that $u_{n+1}-u_{n}\sim\ p\cos(np)$ with $u_{n}=\sin(np)$.
How can I do that? I tried trigonometric formula and Taylor series ...
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Functional Analysis - How is this function expanded?
I am struggling to understand an answer for a good few hours now.
I've got
$\int{ln(1+x^2y')dx}$ with some boundary conditions.
y and h being continuous differentiable functions.
Then for $\delta= S[y+...
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Change of basis and multivariate Taylor series in surface integral
Let $\Gamma$ be a closed smooth surface in $\mathbb{R}^{3}$, and $\mu:\Gamma\rightarrow\mathbb{R}$. We assume $\Gamma$ can be parametrized in $(u,v)$ such that $\mathbf{x}(u,v)\in\Gamma$ for $(u,v)\in ...
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Expansion of log(1-p)/(-p)
I am struggling to prove the following:
\begin{align}
\frac{\log(1-p)}{-p}
=
\sum_{k=1}^{\infty} \frac{1}{k} p^{k-1}
=
1 + \frac{1}{2}p + \frac{1}{3}p^2 + \frac{1}{4}p^3 + \dots
\end{align}
where $p \...
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Help to improve polynomial approximation with example of sine approximation
I found that composition of polynomials has interesting properties for approximation. (1e-20 error for 6 coefficients).
In pseudocode $\ p2=x+ax^3+bx^5$
...
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$g(z) = \sum_{n=0}^{\infty} \frac{g(n)}{(n+1)!} z^n$ with $0 \leq g(n)$?
Im looking for functions $g(z)$ such that
$$g(z) = \sum_{n=0}^{\infty} \frac{g(n)}{(n+1)!} z^n = g(0) + \frac{g(1)}{2} z + \frac{g(2)}{6} z^2 + \frac{g(3)}{24} z^3 + ...$$
and $g(n)$ are all positive ...
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Bounded second derivative also bounds the function
I've been struggling with this problem for a few days:
Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined as
$f(x,y)=xg(y)-yg(x)$, where $g:\mathbb{R}\rightarrow \mathbb{R}$ is
such that $g\in C^2$, $...
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Is there any function in which the Maclaurin series evaluates to having prime numbered powers and factorials? [duplicate]
I am searching for any information or analysis regarding the functions
$$f(x)=\sum_{n=1}^{\infty}\frac{x^{p\left(n\right)}}{\left(p\left(n\right)\right)!}$$
or
$$g(x)=\sum_{n=1}^{\infty}\frac{\left(-1\...
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$f(z) = \sum_{n=0}^{\infty} f(n)^2 z^n$?
Im looking for functions $f(z)$ such that
$f(z) = \sum_{n=0}^{\infty} f(n)^2 z^n = f(0)^2 + f(1)^2 z + f(2)^2 z^2 + f(3)^2 z^3 + ...$
and $f(n)$ are all real.
And I wonder how fast $f(n)$ grows.
I had ...
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Approximating $\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}$ [duplicate]
Consider the series
$$
\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}
$$
How many terms of this series do I need to consider to have an approximation of $\pi$ accurate up to $10$ decimal places (for ...
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Given the function $f$ defined by $f(x, y) = xe^{y} + 1$ develop into a Taylor series
Given the function $f$ defined by $f(x, y) = xe^{y} + 1$:
Develop $f$ into a Taylor series around the point $(1, 0)$
I know how to develop around $(0, 0)$:
$T(x, y) = 1 + x\sum_{k=0}^{\infty} \frac{y^...
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Big-O Notation of Maclaurin Formula for cos x
I am self-studying the book Calculus, a Complete Course By Adams. On the section on Maclaurin formulas for some elementary functions, the formula for $\cos x$ is written as follows with the error in ...
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Cylinder area minimization for constant volume
For this problem, a tin can is being used as an analogy. If no tin is wasted we use the least amount of tin when $h=2r$. However if the bases of the cans were cut out of square tin plates with sides $...
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Taylor series of $\sum_{n=0}^{\infty} \frac{\cos(n^2 x)}{2^n}$
I have the following problem. I must show that the following function $f$ is infinitely differentiable, then find its Taylor series centered at $0$, and the corresponding radius of convergence:
$$f(x) ...
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Minimal distance between zero's $d = D(f(z)) = \inf_{i \neq j} |(z_i - z_j)| s.t. f(z_n) = 0 $?
Let $f(z)$ be a transcendental entire function.
Hence $f$ is not a polynomial.
Assume $f(z)$ has infinitely many zero's $z_n$
$$f(z_n) = 0$$
Lets say that $f(z)$ is given by a taylor series.
Im ...
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A proof that Newton method converges
I am asked to write down the Taylor series for a function $f$ evaluated at $x + h$ in terms
of $f(x)$ and its derivatives evaluated at $x$. Then, to use this result to show that if $x_0$ is an ...
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$n$th term of a Maclaurin series
I am having some confusion about the $n$th term of a Maclaurin series.
For instance, $e^x = 1 + x + \frac{x^2}{2!} + ... + \frac{x^n}{n!}+...$
The general term is $\frac{x^n}{n!}$ but in reality that ...
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Taylor Series Expansion for variance?
I see that the first derivative of variance, wrt $x_i$ is $\frac{2}{N}(x_i - \mu)$. And the second derivative would be simply $\frac{2}{N}$ (beyond this, all subsequent derivatives should be zero.)
So ...
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$\sum_{i=0}^b \frac{n^i}{i!}$ =? [duplicate]
I am trying to solve the summation $\sum_{i=0}^b \frac{n^i}{i!}$. I know that it´s the Taylor exapnsion of $e^n$ as b $\rightarrow \infty$, but I´m having some trouble with the remainder. Is there a ...
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Suppose $\int_0^1 f(x)\space dx =0$, then $\forall t \in [0,1] : |\int_{0}^{t}f(x)\space dx|\leq1$.
Let $f: [0,1] \rightarrow \mathbb{R}$ be differentiable s.t $\forall x \in [0,1]: |f'(x)|\leq8$ .
Suppose $\int_0^1 f(x)\space dx =0$, then $\forall t \in [0,1] : |\int_{0}^{t}f(x)\space dx|\leq1$.
...
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Taylor series approximation usage confusion
Hello Math Exchange!
I'm writing a short justification for gradient descent algorithm (nothing too rigorous, I'm a machine learning student) for my thesis, so in the course of reading up on it, I ...
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Why is $g(c) = 0$ and $g^{(k)}(c) = 0$ for $k < n$ in this proof of Taylor's theorem?
I am reading a proof of Taylor's theorem given in Ross' Elementary Analysis, but I just cant't figure out this one step.
Definition: The remainder of the Taylor series for $f$ about $c$ is
$$
R_n(x) = ...
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Series expansion of $\frac{e^z}{z}$ around $z=-1$ in region $|z+1|<1$.
Basically, I approached this problem using the basic Taylor Series Expansion formula:
$$f(z)=f(z_0) + \frac{f'(z_0)}{1!}(z-z_0)+\frac{f''(z_0)}{2!}(z-z_0)^2+\cdots$$
From there, I got stuck with
$$f(z)...
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Taylor series of same function being different?
I'm trying to expand ln(3+4x) into a taylor series
$\ln(3+4x) = \ln(1+2+4x)$
Let t = 2(1+2x)
$\ln(1+t) \approx t-\frac{t^2}{2} + \frac{t^3}{3} - \frac{t^4}{4} + \frac{t^5}{5}$
So then
$\ln(1+t) \...
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Point in use of Taylor Series to approximate functions in an age with computers?
I hope this doesn't sound too vague or like I'm dismissing the use of Taylor Series entirely, I'm just curious about any proper real-world applications.
Many times Taylor Series are shown-off as a ...
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In what sense, Taylor series at $x=0$ is better than Taylor series at $x=a$, and vice-versa?
In the book `Calculus Early Transcendentals (6th Edition)' by James Stewart, on p. 739, the author writes:
We have two series representations for $e^x$ as follows:
$$e^x=\sum_{n=0}^\infty x^n/n! , \...
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Gradient descent derivation
Background: Regular gradient descent can be written something like $x_{t + 1} = x_t - \eta g_t$, where $g_t$ is the gradient of the function we're trying to optimize.
Problem: If we have a (symmetric, ...
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How to get derivatives from Taylor series [closed]
I recently started learning about Taylor series and getting the derivative is confusing from the sum provided. So for example if I had the Taylor series
$$f(x) = \sum_{k=0}^\infty (-1)^{k+1}\frac{k!}{(...
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Taylor approximation of exponential constraints
I'm currently tackling an optimization problem that involves an exponential constraint, and I'm trying to apply the following technique to transform it into second-order cones. However, I'm struggling ...
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Why does this sum not diverge?
So I'm asking this question in general but with a motivating example:
Say we have a function $$f(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$
So both the top ($x^n$) and bottom ($n!$) of this function grow -...
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A question on the coefficients of the Taylor's series of an entire function
Let $f:\mathbb{C} \to \mathbb{C}$ be defined by $$f(z)=(1-z)e^{\big( z+ \frac{z^2}{2} \big)}=1+ \sum_{n=1}^{\infty} a_nz^n.$$
Then, which of the following is FALSE?
$f'(z)=-z^2e^{\big( z+ \frac{z^2}{...
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Taylor Expansion of the Polynomial in Flajolet’s Fundamental Lemma
I am currently looking at the proof of Flajolet’s Fundamental Lemma. Before I phrase the question, I need to review the definition of $(0,k)$-path and define its weight.
Define $(0,k)$-path as the ...
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For all complex $|z| \neq 1$ : $\frac{z}{1+z+z^{2}+z^{3}+z^{4}} = \sum_{n=0}^{\infty} T_n \frac{z^n}{1+z^n+z^{2n}}$?
Inspired by this one
For all complex $|z| \neq 1$ : $\sum_{n=0}^{\infty} w_n \frac{z^n}{1+z^n+z^{2n}+z^{3n}+z^{4n}} = \sum_{n=0}^{\infty} u_n \frac{z^n}{1+z^n+z^{2n}}$?
It made sense to me, to take ...
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Question on Transforming the Sum of Two Infinite Series into the Sum of Two Exponentials Using Maclaurin Series
I am analyzing the differential equation, $y''-4y=0$.
Its characteristic equation is given as $\lambda^2 - 4 = 0$. Then, we can easily get $\lambda_1=2$ and $\lambda_2=-2$. Accordingly, we finally get ...
- 3,365
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Application of the binomial series: In what point was the Taylor series developed and how?
Let $|\vec{r}| \gg\left|\vec{r^{\prime}}\right|$ and $x:=\frac{\vec{r}^{2}-2 \vec{r^{2}} \cdot \vec{r^{\prime}}}{\vec{r}^{2}} $ be a small number.
$$\frac{1}{\left|\vec{r}-\vec{r^{\prime}}\right|}=\...
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Show that $ \lim_{x \to -\infty} (1 + \sum_{n=1}^{\infty} (\frac{x}{\ln^2(n+1)})^n ) = 0$
Let $x$ be real and define the entire function $f(x)$ as
$$ f(x) = 1 + \sum_{n=1}^{\infty} (\frac{x}{\ln^2(n+1)})^n $$
Now we have that
$$ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (1 + \sum_{...
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$f_c(x) = \sum_{k = 0}^{\infty} a_k x^k =a_0 + a_1x + a_2x^2 + ...$ has no closed form apart from rational functions?
Consider a taylor series with nonzero radius :
$$f_c(x) = \sum_{k = 0}^{\infty} a_k x^k =a_0 + a_1x + a_2x^2 + ...$$
Such that the set/list $a_n$ is a bijection to the set of integers larger than a ...
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1
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Multivariate limit using multivariate Taylor
I'd like to compute the following limit using the multivariate Taylor theorem:
$$
\lim_{(x,y)\to (0,0)} \frac{e^{xy}-1}{x}
$$
I know that this limit is $0$, since
$$
\frac{e^{xy}-1}{x}=y\frac{e^{xy}-1}...
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1
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On the lacunary function $f(z) = \frac{r + f(-1)}{\sqrt 5}$ [closed]
Consider the lacunary series
$$ f(z) = \sum_{n=1}^{\infty} a_n z^n $$
with radius $1$ and natural boundary at the unit circle.
The $a_n$ are real and strict positive, and also
$$\sum_{n=1}^{\infty} ...
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1
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How to prove Taylor's theorem?
Here is a proof of mean value theorem:
Consider a line passing through the points $(a, f(a))$ and $(b, f(b))$. The equation of the line is
$y-f(a) = \displaystyle\frac{f(b)-f(a)}{(b-a)} (x-a)$
or $y = ...
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1
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1
answer
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Expansion of $1/\log\left(\dfrac{f(a)^2 - f(x)^2}{f(a)^2 + f(x)^2}\right)$ around $a$
I want to know what's the order of the second leading term in the expansion of
$1/\log\left(\dfrac{f(a)^2 - f(x)^2}{f(a)^2 + f(x)^2}\right)$ around $a$. For now, I have:
$$1/\log\left(\dfrac{f(a)^2 - ...
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0
answers
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For all complex $|z| \neq 1$ : $\sum_{n=0}^{\infty} w_n \frac{z^n}{1+z^n+z^{2n}+z^{3n}+z^{4n}} = \sum_{n=0}^{\infty} u_n \frac{z^n}{1+z^n+z^{2n}}$?
Ok I am a bit confused.
So here comes a question,
Consider a maclaurin series for $f(z)$
$$f(z) = \sum_{n=0}^{\infty} f_n z^n$$
where $f(z)$ has a radius of exactly $1$.
$f(z)$ may or may not have a ...
- 15.3k
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votes
0
answers
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Matrix and Taylor Expansion of a Finite Continued Function
I noticed a pattern between matrix and Taylor series of a finite continued fraction function. However, I don't know how to prove it or why they are related.
Let
$$
f_{1}(z)=\frac{1}{-z-1}
$$
$$
f_{2}(...
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