Tagged Questions
1
vote
1answer
27 views
Finding power series for $f(x) = \frac{4x+53}{x^2-x-30}$
Given $f(x) = \dfrac{4x+53}{x^2-x-30}$, display it as a power series and find the radius of convergence. then calculate $f^{(20)}(0)$
So what I did was look at the Taylor Series Formula:
$$f(x) = ...
0
votes
2answers
67 views
Taylor series of $f(x)=\frac {e^x-1}{x}$
I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions.
How to simplify the function so that it can be expanded more easily?
1
vote
3answers
59 views
Taylor Polynomial for $x^{1/3}$
a. Compute the Taylor polynomial $T_3(x)$ for the function $(x)^{1/3}$ around the point $x=1$.
b. Compute an error bound for the above approximation at $x = 1.3$.
I'm having trouble figuring ...
5
votes
0answers
65 views
Maclaurin series of $f(x)=\sinh(1/x)$?
As we know the formula of Maclaurin series for $f(x) = \sinh(x)$ is $f(x)=x+x^3/3! + x^5/5!+\ldots$
Could anyone tell me what is the Maclaurin series of $f(x)=\sinh(1/x)$?
1
vote
0answers
25 views
Binomial Expansion problem error
I tried solving this question but failed.
a) Expand $(1+2x)^{1/4}$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each term as far as possible.
b) By substituting ...
0
votes
1answer
70 views
Finding the error of the Taylor expansion of $\log(1 + x)$
The questions is as defined below.
Let $f(x)= \log(1+x)$. Show that the Taylor remainder $R_{0,k}(x)$, defined by $$R_{a,k}(x)= f(a+x) - P_{a,k}(x) = f(a+h) -\sum_{j=0}^{k} \frac ...
2
votes
2answers
58 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 ...
0
votes
2answers
67 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 ...
1
vote
1answer
33 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 ...
4
votes
3answers
56 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 ...
6
votes
3answers
62 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 ...
2
votes
2answers
63 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 ...
0
votes
1answer
27 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 ...
1
vote
0answers
78 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 ...
0
votes
1answer
29 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 ...
2
votes
1answer
51 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 ...
2
votes
1answer
56 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 ...
1
vote
1answer
54 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 ...
0
votes
1answer
65 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?
2
votes
1answer
54 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 $x=1$
0
votes
2answers
46 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 ...
1
vote
1answer
40 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
$$\dfrac{1 - ...
0
votes
3answers
95 views
$k$th term of Taylor series of function $f(x) = 3x^3-2x+4$
I'm having trouble solving this question. I have all the values right but can't figure out a way to find the $k$th term. My Taylor series values up to that point are ...
1
vote
1answer
50 views
Why do Maclaurin series approximate a function for negative domain values?
A common analogy used as an intuitive explanation for a Maclaurin series is that of a car. If you know the position, velocity, acceleration, jerk etc. of a car at time zero, you are able to predict ...
1
vote
0answers
24 views
Inequality of Partial Taylor Series
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} \frac{x^k}{k!} ...
1
vote
4answers
141 views
MacLaurin series of $\ln(1-x^2)$
The MacLaurin series for $\ln(1 + x)$ is obtained from the series for $\frac{1}{1 + x}$
by integration. Use this and appropriate substitutions to obtain the MacLaurin series for $\ln(1-x^2)$.
...
2
votes
2answers
126 views
Finding the Taylor series of $\log x$ at $x=1$ and $2$
How do I find the Taylor series of the following functions:
$f(x)=\log(x)\ (x>0)$ at the point $x=1$
$g(x)=\log(x)\ (x>0)$ at the point $x=2$
Help greatly appreciated
1
vote
1answer
53 views
Curvature via hessian in Taylor expansion
In the case of a univariate function, the smaller the second derivative in its Taylor expansion, the smaller is the curvature of the univariate function.
Now, how is the curvature of the function ...
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 ...
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 ...
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 $3$ ...
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
57 views
Having trouble showing that these series are the same.
$$\frac{\sqrt{2}}{2}\sum \limits_{n=0}^{\infty} (-1)^{\tfrac{n(n+1)}{2}+1}\frac{(x-\pi/4)^n}{n!} $$
$$= \frac{\sqrt{2}}{2}\sum \limits_{n=0}^{\infty} ...
1
vote
1answer
52 views
Composition Taylor Series
Is there any theorem that specifies when we are allowed to compose the taylor series of two functions? Does it have a name?
Thanks.
0
votes
1answer
66 views
Taylor Polynomial Proof
I am going over a previous year's test and I have no idea how to approach this question. If anyone could please help. Let $g(x)=e^{x^2}$.
1
vote
2answers
88 views
Finding the Maclaurin series
Find the Maclaurin series for $f(x)=(x^2+4)e^{2x}$ and use it to calculate the 1000th derivative of $f(x)$ at $x=0$.
Is it possible to just find the Maclaurin series for $e^{2x}$ and then multiply it ...
1
vote
2answers
53 views
Question about taylor expansion
If I was given a function which its derivative is bounded for every $x>0$ (means: $|f'(x)|\le M$), How can I prove that $\lim_{x\to\infty}\frac{f(x)}{x^2}=0$?
1
vote
1answer
45 views
Taylor polynomials expansion with substitution
I am working on some practice exercises on Taylor Polynomial and came across this problem:
Find the third order Taylor polynomial of $f(x,y)=x + \cos(\pi y) + x\log(y)$ based at $a=(3,1).$
In the ...
0
votes
1answer
34 views
Polynomial of degree four of $f(x)=\sqrt{x}$
Given $f(x)=\sqrt{x}$ Find a polynomial $P(x)$ of degree three such that $P^{(k)}(4)=f^{(k)}(4)$ for $k=0,1,2,3,4$.
I know this has to do with Extended mean value theorem, or, Taylor Formula. ...
2
votes
2answers
66 views
Taylor expansion with change of variables question.
Find the Taylor polynomial of order 3 of
$$f(x,y) = (x - 1)^{2} + \sin(\pi y) + x \ln(y)$$
based at $(x,y) = (2,1)$.
So I'm really lazy and don't want to take the derivative of that, so let ...
2
votes
2answers
48 views
Find the Taylor series of $\frac{1}{x+1} $ at $x=2$
This is what I did:
$\begin{align*}
f(x)&=&(x+1)^{-1}\\
f'(x)&=&-(x+1)^{-2}\\
f''(x)&=&2(x+1)^{-3}\\
f'''(x)&=&-6(x+1)^{-4}\\
f''''(x)&=&24(x+1)^{-5}\\
...
1
vote
1answer
83 views
Taylor Expansion of the 1/2th Derivative
In trying to solve the problem $\sqrt D f(x)=g(x)$ I tried to expand the derivative as a Taylor series, and have encountered a lot of problems. Is there some reason that this shouldn't be possible? ...
4
votes
1answer
82 views
A sort of “Taylor expansion” of a power series
I have the following question. Suppose $$f(x):=\sum_{i=0}^{\infty}c_ix^i$$ is a power series that converges for $|x|<1 + \epsilon$, for some $\epsilon >0$, where $x\in\mathbb{C}$. I can then ...
1
vote
1answer
29 views
$R_{n}\left ( x \right )$ for $ f\left ( x \right) = \cos\left ( 2\cdot x \right ) $?
How can I define
$R_{n}\left ( x \right )$ for
$$ f\left ( x \right) = \cos\left ( 2\cdot x \right ) $$
I found taylor expansion for $cos2x$.What should I do after that? My problem is I dont know if ...
7
votes
4answers
283 views
Using Taylor's Theorem to show that $\ln(1 + x^2) \leq x^2$
Can we show that if $\operatorname{abs}(x) \lt 1$, then $$\ln(1+x^2) \leq x^2\;,$$
using Taylor's Theorem?
I am thinking of expanding it about $x=0$ but I got something like
$$f(x) = -x^2 + ...
1
vote
1answer
85 views
limit of the error in approximating definite integral with midpoint rule
I want to calculate $\lim_{n \rightarrow \infty} n^2 |\int_{[0,1]}f(x)-I_n(x)|$ where $I_n$ is the integral approximation by midpoint rule:
$I_n=\frac{1}{n}\sum_{k=1}^nf(c_k)$ and $c_k$ is the point ...
