0
votes
2answers
45 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
1
vote
1answer
42 views

Find the Taylor polynomial of degree 4 for cos(x), for x near 0

I am self studying calculus and I need help solving a Taylor Series problem. 1a) Find the Taylor polynomial of degree 4 for cos(x), for x near 0: I think the answer would be: ...
0
votes
1answer
13 views

Taylor Polynomial Variable Question

When you have a polynomial that you set your function equal to in the taylor polynomial (centered around $x = a$) $$function = c_0+c_1(x-a)+c_2 (x-a)^2+...$$ why is your variable $(x-a)$. Oddly ...
0
votes
1answer
21 views

Reference of the expansion of square root polynomials

What is the reference of the formulation given below by Robert israel, please inform me.. Given an even-degree polynomial $$P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_0 = x^{2n} (a_{2n} + ...
2
votes
1answer
44 views

Taylor expansion of polynomial

Intuitively, I would expect the Taylor expansion around $x_0$ of a polynomial in $(x-x_0)$ to be identical to the polynomial. However, I cannot seem to show that/whether this is the case: For a ...
2
votes
3answers
59 views

Proof that Polynomials Form a Basis

I'm not even sure this is a true statement, but can someone prove that the polynomials for a basis for continuous functions? This seems to be motivation for Taylor series, and several of the ...
1
vote
1answer
78 views

Taylor Polynomials, Why only Integer Powers?

So It seems that the definition of polynomial is that is is raised to an integer power, but why is this necessary? My question mainly arises from a proof of the solution to the Hydrogen atom in ...
4
votes
3answers
106 views

What is the relationship between saying “a Taylor series converges for all $x$” and “a Taylor series converges to a function, f(x)”

Given the following Taylor series: $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}- \dots$ We know that: It converges for all of $x$ It converges to the function $\cos x$ The ...
2
votes
0answers
52 views

Approximating polynomial of higher degree

Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values ...
0
votes
1answer
75 views

Finding a generating function for a pattern

I was working on this projecteuler.com problem, and I was very interested by the premise. Essentially, given n terms, find an ...
1
vote
1answer
47 views

Estimate the degree of a Taylor Polynomial using its Error Term

In my 2nd year studying Maths at Uni and revising for a Numerical Analysis final exam. We're given 1 past paper but no solutions, and I can't answer this question: Use the error term of a Taylor ...
1
vote
3answers
88 views

$\frac{1}{1-x}$ series expansion

How do I know that the expression: $$\frac{1}{1-x}$$ Is equal to the infinite sum: $$-\left(\frac{1}{x}\right)-\left(\frac{1}{x}\right)^2-\left(\frac{1}{x}\right)^3-\left(\frac{1}{x}\right)^4+...$$ ...
1
vote
0answers
70 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
2answers
67 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
2answers
2k views

Power expansion for the square root of an even degree polynomial

I am reading an article from 1936 with something that looks like an easy way to solve Riccati equations with variable coefficients as nice polynomials.The link is : ...
1
vote
1answer
84 views

How to compute tangent of a discretized curve

I have a discretized curve defined by a 2D matrix $M$ where $M(i,j)=1$ means the point $(i,j)$ is on the curve. For each of these points, I want to calculate its tangent vector by fitting a polynomial ...
2
votes
3answers
143 views

Precision with Taylor Expansions

when you take a 1st order taylor expansion of a function, so: $$f(a) + f'(a)(x-a)$$ does that mean that if the result is only accurate to one decimal place? so for a value a.bcd, d would be the ...
1
vote
2answers
478 views

Why do we need Taylor polynomials?

This question doubles as "Is my understanding of what a Taylor polynomial is for, correct?" but In order to write out a Taylor polynomial for a function, which we will use to approximate said function ...
5
votes
5answers
193 views

Can every polynomial be written as $a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 +\ldots + a_n (x-x_0)^n$

Can every polynomial of degree $n$ be written as $a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \ldots + a_n (x-x_0)^n$ while $x_0$ is an arbitrary but given real number and all $a_k$ can be freely chosen? Is ...
0
votes
1answer
308 views

a multivariate quadratic function

Assume a vector-valued function, for example ${\bf f}=(f_1, f_2)$, where $$f_1(x,y)= x^2+3xy$$ $$f_2(x,y)= 2xy+y^2$$ (here f is column vector, x, y are variables) Assume that each $f_i$ is a ...
4
votes
1answer
739 views

Remainder term of Lagrange Interpolation Polynomial

Suppose $x_0,x_1,\ldots,x_n$ are $n+1$ distinct numbers in the interval $[a,b]$ and $f\in C^{n+1}[a,b]$. Then for each $x$ in $[a,b]$, there is a number $\xi$ in $(a,b)$ such that $$f(x) = P(x) + ...
11
votes
2answers
402 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 $x = 0$. ...
2
votes
3answers
537 views

Do all polynomials with order $> 1$ go to $\pm$ infinity?

Background As background, I have found that taylor expansion provides poor estimates of a function at extreme parameter values. Indeed, the approximation at extreme values can get worse (more rapid ...
8
votes
2answers
1k views

Taylor series of a polynomial

Given a polynomial $y=C_0+C_1 x+C_2 x^2+C_3 x^3 + \ldots$ of some order $N$, I can easily calculate the polynomial of reduced order $M$ by taking only the first $M+1$ terms. This is equivalent to ...