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.
2
votes
3answers
31 views
Finding Maclaurin series of a given function
For function:
\begin{align}
f(x) = \frac{x}{1+2x}
\end{align}
Can be written as:
\begin{align}
f(x)=x\bigg(\frac{1}{1+2x}\bigg)\tag{1}
\newline = x\sum_{n=0}^\infty (-1)^n2^nx^n\tag{2}
\newline = ...
0
votes
3answers
65 views
Proof by using taylor series
So, everyone that took Single Variable Calculus (calc 1) should be familiar with Taylor Series.
Now, I have a question:
How do I show that:
$$\log(2)=\sum^{\infty}_{n=1}(-1)^{n+1}\frac{1}{n}$$
and ...
2
votes
0answers
167 views
power series of arcsin(x) centered at x = 0
I am trying to prove that the Taylor expansion of $\arcsin(x) = \sum\limits_{n=0}^\infty \cfrac{(2n!)x^{2n+1}}{(2^nn!)^2(2n+1)}$.
Sorry about the notation, I'm not sure what syntax to use. S stands ...
3
votes
1answer
53 views
Intuition behind 2nd order approximation, help please
I know how to apply the formula for Taylor Expansions. But what I want to understand is the intuition. Let me explain with the following example:
If $y=x^5$ its 1st derivative is $5x^4$ and its 2nd ...
0
votes
2answers
64 views
Taylor's theorem for $|x|<1$ for $\sqrt{1+x}$?
I'm trying to do a Taylor expansion on $\sqrt{1+x}$ for $|x|<1$ but I'm not sure how to proceed after finding the derivatives. I'd understand how to do it if it were centered at $a$, but the ...
1
vote
1answer
69 views
Adding two Power series or Maclaurin sums together and their radius of convergence
Say you have two power series. One of them has ROC of 2, and the other one has an ROC of 4. If you add the two series together is the ROC ALWAYS the lesser ROC? It seems to be a trend I've noticed, ...
0
votes
0answers
40 views
Taylors Inequality to evaluate $f(x) = x\sin(x)$ when $a = 0$ and $-1\le x\le1$
Trying to calculate the error of this function when you use a Taylor expansion to degree 4.
I keep getting $.039$ when the answer in the back of the book is $.042$.
I take the fifth derivative of ...
1
vote
0answers
32 views
Why the ith coefficient of $|\lambda I-A|$ is the sum of all $i$-th order principle minors of $A$?
I come across a theorem that
$f(\lambda )=|\lambda I-A|$, which equals to $\lambda ^{n}-a_{1}\lambda
^{n-1}+\alpha _{2}\lambda ^{n-2}-...(-1)^{n}a_{n}$
where $a_{i}$ is the sums of all ith order ...
1
vote
1answer
82 views
Radius of convergence of Maclaurin series for $\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$
What is the radius of convergence of the Taylor series about $z=0$ for $h(z)=\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$?
Here's a plot
...
1
vote
2answers
32 views
Lagrange remainder to approximate $3^{2.1}$ less than 0.1
How do I solve this problem:
Use the appropriate Taylor polynomial $P_n(x,c)$ to estimate $3^{2.1}$ with error less than $0.1$, given $\ln 3$ is about $1.099$.
I understand that the remainder ...
1
vote
2answers
55 views
Question about Maclaurin Series for $\cos x$
I understand how to get the proper maclaurin series representation for $\cos x$, but I'm having trouble understanding the following part conceptually:
I get $\cos x$ as $\sum_{n=0}^\infty ...
1
vote
1answer
41 views
Find taylor polynomial that approximates e^x with accuracy at least 1.
Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$.
I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
6
votes
0answers
89 views
Function $f(x)$, such that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$
Consider a function $f(x)$. Define Taylor series $\sum_{n=0}^{\infty} f(n) x^n$. Is there such a function, other than constant $0$, that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$?
The Taylor series of ...
10
votes
1answer
253 views
Old versus New enunciation of Taylor's Theorem.
I am studying from Spivak' Calculus, and he states Taylor's Theorem as follows:
THEOREM
Let $f',\cdots,f^{(n+1)}$ be defined on $[a,x]$ and let $R_{n,a}(x)$ be defined by
...
0
votes
1answer
323 views
Laurent Expansion of $\frac{1}{z(z-i)^2}$
How can we compute the Laurent expansion of
$$f(z)=\frac{1}{z(z-i)^2}$$
about $0$ in $\Omega=B_1(0)\setminus\{0 \}$?
Thoughts:
By partial fractions we have that
...
0
votes
1answer
141 views
Laurent Series and Taylor Series
I am trying to find the Laurent series of $\dfrac{1}{(1+x)^3}$; would this be the same as finding the Maclaurin series for the same function?
3
votes
2answers
103 views
How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?
I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
2
votes
2answers
121 views
Confused by Laurent series
A typical problem related to Laurent series is this:
For the function $\frac 1{(z-1)(z-2)}$, find the Laurent series expansion in the following regions: $\\(a) |z|<1, \\ (b) 1<|z|<2, ...
1
vote
2answers
55 views
4-th derivative of $(1+x+x^2) / (1-x+x^2) $ using Taylor polynomial for $1/(1-x)$
Using $n$-th Taylor polynomial for $f_1(x)=\frac{1}{1-x}$ with center in $0$, find $4$-th derivative of $f_2(x)=\frac{1+x+x^2}{1-x+x^2}$ in the point $0$ without calculating it's $1$,$2$ or $3$ ...
2
votes
1answer
53 views
Confused over analytic functions, point convergence of power series
It is well-known that a power series sums to a function that is analytic at every point inside its circle of convergence and that conversely, if a function is analytic on an open disc then its Taylor ...
2
votes
1answer
33 views
Two-point Taylor expansion with one assymptotic point?
According to this paper, a two-point Taylor expansion can be definied like this:
$$\text{Let }f\left(z\right)\text{ be an analytic function and }z_1 \text{and }z_2\in \mathbb{C}, z_1\neq ...
0
votes
0answers
34 views
Taylor expansion of an integral in spherical co-ordinates
I've some difficulty deriving this equation from jackson electrodynamics (The equation after 1.30)
$\nabla^2 \Phi_a\left({\textbf{x}}\right)=-\frac{1}{\epsilon_0}\int_{0}^{R} ...
3
votes
2answers
38 views
Range of the sine function
It is obvious from the definition of $f(x)=\sin(x)$ using the unit circle of radius $1$ that the range of that function is the set $[-1,1]$. But also there are approaches where the sine is defined ...
2
votes
2answers
48 views
Solving limit by substituting a power series
I dont understand why I am getting 2 and the textbook says it is -2.
$$\lim_{x\to 0} \frac{1-e^x}{\sqrt{1+x}-1}$$
I subbed the power series for $e^x$ and $(1+x)^{1/2}$ then got rid of the $1$ on top ...
2
votes
2answers
102 views
A Question On Euler's Proof Of the Basel Problem
I've studied the proof that Euler gave for the famous Basel Problem, and it would seem that while it is technically correct, he does not justify all of his steps properly. Namely, he assumes that
...
0
votes
1answer
37 views
Calculating a derivative from a Maclaurin series
I'm attempting to find $f^{(100)}(x)$ for $f(x)=\frac{1}{1+x^2}$. The Maclaurin series is $\sum\limits_{n=0}^{\infty}(-1)^nx^{2n}$.
I figure that $(-1)^nx^{2n}=\frac{f^{(n)}(0)\;x^n}{n!}$. Setting ...
2
votes
2answers
57 views
Having trouble showing that these series are the same.
$$\frac{\sqrt{2}}{2}\sum \limits_{n=0}^{\infty} (-1)^{\tfrac{n(n+1)}{2}+1}\frac{(x-\pi/4)^n}{n!} $$
$$= \frac{\sqrt{2}}{2}\sum \limits_{n=0}^{\infty} ...
2
votes
3answers
85 views
Check series convergence using Taylor series
Kinda stuck in here too, some kind of help would be greatly appreciated!
Using Taylor expansion check if series :
$$\sum_{n=1}^{\infty}\sqrt{n}(\arctan{(n+1)} - \arctan{n})$$
converges.
Thanks in ...
0
votes
1answer
40 views
Taylor expansion, series convergence
Kinda stuck in here too, would like to get some help!
Using Taylor expansion check if series :
$$ \sum_{n=1}^{\infty}\frac{1}{n}\sqrt{e^\frac{1}{n} - e^\frac{1}{n+1}} $$
converges.
How do I ...
2
votes
4answers
54 views
For $x > -1$ proof that $ \arctan x + \arctan\frac{1-x}{1+x} = \frac{\pi}{4} $
For $x > -1$ proof that $\arctan x + \arctan\dfrac{1-x}{1+x} = \dfrac{\pi}{4} $
I have no idea how to approach this, some kind of help would be greatly appreciated!
edit: Thank you all!
1
vote
0answers
26 views
Is there a way to check if my Taylor Expansion is correct?
I have an exam later and I need to do Taylor expansions of functions. I have questions like:
Consider the map $F:\mathbb{R}^2_x \rightarrow \mathbb{R}^2_y$, given by the equations
$$y_1 = ...
1
vote
3answers
51 views
Series expansion $(1-\cos{x})^{-1}$
How do i get the series expansion $(1-\cos{x})^{-1} = \frac{2}{x^2}+\frac{1}{6}+\frac{x^2}{120}+o(x^4)$ ?
0
votes
0answers
21 views
is this valid + Taylor series
is the following taylor series withlogarithms powers for the m power of x +
$ x^{1/k} =1+ \sum_{n=0}^{\infty} \frac{log^{n}(x)}{n! k^{n}} $
at least for x >1 , so if we truncate up to the term 1000 ...
1
vote
3answers
49 views
Precise differences in meaning of Power Series, Taylor Series
Being an physicist/artist, not a real mathematician, I often toss around the terms "Taylor Series" and "Power Series" without any concern. Are these terms be considered interchangeable by ...
3
votes
2answers
83 views
Infinite (Taylor) Series
What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$?
It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$
What power is $(-1)$ supposed to be raised ...
1
vote
1answer
52 views
Composition Taylor Series
Is there any theorem that specifies when we are allowed to compose the taylor series of two functions? Does it have a name?
Thanks.
0
votes
2answers
173 views
Multivariate Taylor Expansion
I am in confidence with Taylor expansion of function $f\colon R \to R$, but I when my professor started to use higher order derivatives and multivariate Taylor expansion of $f\colon R^n \to R$ and ...
0
votes
1answer
67 views
Taylor Polynomial Proof
I am going over a previous year's test and I have no idea how to approach this question. If anyone could please help. Let $g(x)=e^{x^2}$.
1
vote
2answers
88 views
Finding the Maclaurin series
Find the Maclaurin series for $f(x)=(x^2+4)e^{2x}$ and use it to calculate the 1000th derivative of $f(x)$ at $x=0$.
Is it possible to just find the Maclaurin series for $e^{2x}$ and then multiply it ...
2
votes
0answers
70 views
Series expansion of a series
I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. ...
3
votes
3answers
217 views
Finding the Taylor series expansion of $f(z)=\frac{e^{z}-1}{z}$ around $0$
Find the Taylor series expansion of $f(z)=\displaystyle\frac{e^{z}-1}{z}$ around $0$.
I have no idea where to start.
1
vote
1answer
149 views
A proof of the the second derivative test?
Suppose $f\in C^3$ in some ball centered at a, where $a\in \Bbb{R}^2$,and $\nabla f=0$ at a, but not all second derivatives of $f$ are zero at a. Show how can local maximums local minimums or neither ...
1
vote
2answers
53 views
Question about taylor expansion
If I was given a function which its derivative is bounded for every $x>0$ (means: $|f'(x)|\le M$), How can I prove that $\lim_{x\to\infty}\frac{f(x)}{x^2}=0$?
1
vote
1answer
45 views
Taylor polynomials expansion with substitution
I am working on some practice exercises on Taylor Polynomial and came across this problem:
Find the third order Taylor polynomial of $f(x,y)=x + \cos(\pi y) + x\log(y)$ based at $a=(3,1).$
In the ...
0
votes
1answer
34 views
Polynomial of degree four of $f(x)=\sqrt{x}$
Given $f(x)=\sqrt{x}$ Find a polynomial $P(x)$ of degree three such that $P^{(k)}(4)=f^{(k)}(4)$ for $k=0,1,2,3,4$.
I know this has to do with Extended mean value theorem, or, Taylor Formula. ...
1
vote
1answer
65 views
Exponential as power series
Is there a function that does not depend on $a$ such that $\sum_{x=1}^\infty \frac{a^x}{x!}f(x) = \mathrm e^{-a}$?
Just to be clear, the summation starting from 1 is intentional, otherwise the ...
1
vote
0answers
74 views
Consistency order of backward Euler method
How can I proof that backward Euler method has consistency order 1?
Implicit function theorem states that for a sufficiently small $h$,
$$
\vec{y}_1 = \vec{y}_0 + h f(t_1,\vec{y}_1)
$$
has a unique ...
1
vote
2answers
179 views
Stirling's Formula - Comparison Test Method
The following question concerns the convergence of Stirling's Approximation for $n!$
I have $r_n = \frac{\sqrt{n}}{n!}(\frac{n}{e})^n$.
I have expressed ...
2
votes
2answers
66 views
Taylor expansion with change of variables question.
Find the Taylor polynomial of order 3 of
$$f(x,y) = (x - 1)^{2} + \sin(\pi y) + x \ln(y)$$
based at $(x,y) = (2,1)$.
So I'm really lazy and don't want to take the derivative of that, so let ...
3
votes
2answers
53 views
Prove that $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$ where $|R_8(x)|\leq \frac{x^8}{8!}$
Prove that
$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$$
where $|R_8(x)|\leq \frac{x^8}{8!}$
