0
votes
1answer
28 views

Zeros of polynomials and power series

Consider $k_i \in \{-1,1\}$ for every $i \in \mathbb{N}$ and consider the family of polynomials $P$ of the form $$\mathop{\sum}\limits_{i=0}^n k_it^i;$$ and the family of power series $S$ of the form ...
1
vote
2answers
72 views

Is there any closed form for this series?

It's a power series that I found during the computation for my research. \begin{equation*} \sum_{k=0}^n \binom{n}{k}\frac{n!}{(n-k)!}x^{n-k}(-1)^k. \end{equation*} Without the annoying term of ...
1
vote
2answers
43 views

polynomial series and root multiplicity

Excuse me, because I know this is a double post but I can't for the life of me find the original post. Given a sequence $(a_n)$, one can construct a polynomial of the form ...
4
votes
1answer
77 views

Approximating $e^{inx}$ by polynomials

Show that every function $e^{inx}$ can be uniformly approximated on $[-\pi,\pi]$ by polynomials in $x$. Using the power series expansion, ...
0
votes
0answers
23 views

Simplifying series with unidentified polynomials

I have encountered the following sum: $\sum_{n=0}^{\infty}\frac{t^{n}}{n!} \left(\frac{d^{2}}{dx dy}\right)^{n}e^{txy},$ where $t$, $x$ and $y$ are real numbers. The differentiation results in some ...
4
votes
1answer
118 views

When does an analytic function grow faster than a polynomial?

Suppose $f$ is an analytic function with power series expansion $f(z)=\sum_{n=0}^{\infty} a_nz^n$, and $p = \sum_{n=0}^{d}b_nz^n$ is a polynomial. If $f$ is a polynomial of degree larger than $d$, ...
1
vote
1answer
57 views

Writing a sum as a fraction

Express $$\sum^{20}_{i=2}f(x)^i$$where $$f(x)=\sum_{i\geq 1}2^{i-1}x^{3i}$$ as a fraction of polynomials $p(x)/q(x)$ and simplify as much as possible. Hmm. How to do it? Wolfram is really stupid on ...
1
vote
1answer
116 views

Determining power series for $\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$

I'm looking for the power series for $f(x)=\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$ My approach: the given function is a combination of two problems. first i made some transformations, so the function looks ...
1
vote
3answers
46 views

Rational polynomial from coefficents

Given two polynomials $$ p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \\ q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n} $$ And the series expansion from their rational polynomial $$ ...
0
votes
2answers
71 views

Substituting a binomial into the infinite geometric series formula

In this case, regarding series $\frac1{2+x}$: Can you use $(1-x)$ as the common ratio instead of factoring out a $\frac12$ and using $-\frac{x}2$ [as the common ratio]. WolframAlpha says that ...
0
votes
2answers
123 views

Factoring polynomials over a power series ring

Could anyone tell me why $g(x, y) = x^3 - y^2$ is irreducible in $\mathbb C[[x]][y]$ while $f(x, y) = x^3 - x^2 + y^2$ is not?
0
votes
2answers
102 views

How to efficiently compute the coefficients in a bi-binomial expansion?

Is there a computationally efficient way of calculating the coefficients of the polynomial expansion of expressions like $(1+x^a)^m(1+x^b)^n$ for arbitrary positive integers $m,n,a,b$ (and especially ...
1
vote
2answers
475 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 ...
6
votes
1answer
883 views

prove that the entire function f is a polynomial.

Suppose that $f$ is an entire function, and that in every power series $f(z)=\sum_{n=0}^{\infty} c_{n}(z-a)^n$ at least one coefficient is 0. Prove that $f$ is a polnomial. Hint: ...
3
votes
2answers
162 views

Trouble with representing power series as polynomials.

I am a math student trying to wrap my head around complex analysis through self-study. I am using Complex Analysis by Serge Lang, but I find myself struggling with some of his power series ...
3
votes
1answer
252 views

Is this polynomial positive?

Let $p\geq 2$, and $p$ is not a half odd integer. $t\in R$. Is the following polynomial positive: $$ T_k(t)=\left(\frac ...
2
votes
1answer
101 views

Coefficient signs in the sum of successive powers of a polynomial

I'm searching for some structure in the sign variation of the coefficients of: $$P = \sum_{i>0} p^i\enspace,$$ for some polynomial $p \in \mathbb{Z}\langle x\rangle$ with no constant term. I'm ...
5
votes
3answers
141 views

Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?

The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...
5
votes
2answers
183 views

How to transform the factored form of $\sin(x)$?

We know $\sin(x)=0$ has solutions $0,\pm\pi,\pm2\pi,\pm3\pi,\dots$. So $\sin(x)$, if interpreted as a polynomial, could be written as: $a_0x^0+a_1x^1+a_2x^2+\cdots$ and we know this polynomial too: ...
0
votes
1answer
240 views

Sum of the polynomial roots raised to a power. How to prove?

Problem: If we have a polynomial $f$ with a derivative $f\,'$ and quotient $q$ function defined as: $$q(x)=\sum_{i=1}^{\infty}a_ix^{-i}=\frac{f\,'(x)}{f(x)},$$ and the roots of $f$ are ...
3
votes
2answers
432 views

How to compute coefficients in Trinomial triangle at specific position?

I need to compute coefficients of $i$-th power of $x$ from simplifying of $$(x^2 + x + 1)^n$$ From that site i know about trinomial triangle. But how can compute coeficients of $i$-th element at ...