1
vote
1answer
25 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
0
votes
2answers
53 views

Prove that no polynomial function represents a given sequence

I recently encountered the following question: How to find the $n$th term of the sequence $2,3,6,7,14,15,30,\dots$? I replied to that post, and gave the following answer: $S_n = ...
12
votes
1answer
144 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
0
votes
0answers
42 views

Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
0
votes
1answer
31 views

Why does the interpolation error go to zero if we increase the number of sampling points?

This question is motivated by polynomial interpolation. We know that for $f\in C^{n+1}[a,b]$ and $a=x_0<\dots<x_n=b$ holds $$\| f - p_n \|_\infty \leq \frac{1}{(n+1)!} \| f^{(n+1)} \|_\infty ...
0
votes
0answers
20 views

What is the series expansion of this polynomial to the negative power

$$(1+x+x^2/2 +5x^3/18 +25x^4/144)^{-2}$$ How would I go about finding the series expansion of this? I know how to use the Taylor expansion, but not how the negative power would affect the answer.
0
votes
1answer
13 views

Harmonic sum of sequence of real numbers

if $\left \{ x_n \right \}$ is a sequence of real numbers such that \begin{equation} x_1=2 \tag{1} \end{equation} \begin{equation} x_{n+1}=x_n^2-x_n+1, \,\,\forall n=1,2,3\cdots\ \tag{2} ...
2
votes
1answer
51 views

What is this sequence of polynomials?

NovaDenizen says the polynomial sequence i wanted to know about has these two recurrence relations (1) $p_n(x+1) = \sum_{i=0}^{n} (x+1)^{n-i}p_i(x)$ (2) $p_{n+1}(x) = \sum_{i=1}^{x} ip_n(i)$ == i ...
3
votes
4answers
40 views

Finding the Remainder

Given the polynomials $$P(x) = nx^n+(n-1)x^{n-1}+(n-2)x^{n-2}+\cdots+x+1$$ and $$Q(x)=x(x-1)^2$$ find the remainder of the division $\dfrac{P (x)}{Q (x)}$.
0
votes
1answer
36 views

Uniqueness of infinite polynomial functions

This is not a homework question, it is just something I was wondering about. Suppose we have 2 sequences of real numbers, ${a_i}$ and ${b_i}$, and their respective polynomials $A(x) = ...
4
votes
1answer
67 views

Show that $f(z)=\sum_{n= 0}^{+\infty}a_n z^n$ is a polynomial

Let $f(z)=\sum_{n= 0}^{+\infty}a_n z^n$, the radius of convergence $\ge 1$. For all $n,\quad a_n\in \mathbb{Z}$ and $f$ is bounded the open unit disk. Show that $f$ is a polynomial. My ...
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 ...
1
vote
2answers
48 views

Taylor Series - Write $1 + 2x -x^2 + 5x^3 - x^4$ at powers of $(x-1) $

Exercise: Write the polynomial $1 + 2x -x^2 + 5x^3 - x^4$ at powers of $(x-1)$. I presume this exercise is solved using Taylor Series, since it belongs to that chapter, but have no idea how to solve ...
0
votes
4answers
108 views

1, 3, 6, what is the next number of the sequence?

I've heard (and believed even without proof) that given any finite sequence there is more than one formula for which the same first inputs give the same first outputs. Given that: f(1)=1 f(2)=3 ...
1
vote
2answers
32 views

Polynomials through successive differences

Let $h_0:\Bbb{N}\rightarrow\Bbb{N}$ be any function. Define recursively, for $m>0$, $$h_{r+1}(m)=h_r(m)-h_r(m-1).$$ Suppose that for some $k>0$ we have $h_k(m)\equiv d$ constant. Is this ...
1
vote
3answers
54 views

A question on Arithmetic Progressions is given in the picture below..

What is the sum of an arithmetic progression whose first term is $a$, the second term is $b$, and the last term is $c$? A. $\dfrac{(b+c-2a)(a+c)}{2(b-a)}$ B. $\dfrac{(b+c+a)(a+c)}{2b-a}$ C. ...
4
votes
1answer
55 views

Existence of a sequence that converges to a polynomial

Let $P\in \mathbb{R}[X]$ Is there a norm $\|\cdot\|$ such that the sequence $(X^n)_{n\in \mathbb{N}}$ which converges to $P$? Thank you
7
votes
2answers
120 views

How to prove that $b_{2008}\neq 0$

Let the polynomial $f$ be defined as $$f(x)=a_{m}x^m+a_{m-1}x^{m-1}+\cdots+a_{1}x+a_{0}, \qquad a_{i}\in \Bbb Z \ (i=0,1,2,\cdots,m), \ a_{i}\neq 0.$$ Define the sequence $\{b_{n}\}$ as ...
1
vote
1answer
196 views

In EMI calculations, how to calculate “Rate” if EMI, Principal and Time are given

In EMI (Equated Monthly Instalments) calculations, the inputs are- Principal-P, Rate-r, and ...
0
votes
3answers
50 views

Finding the coefficient of a variable in a polynomial

Find the coefficient of $x^8$ in the polynomial $(x-1)(x-2)\cdots\cdots(x-10)$. How do I approach such problems?
1
vote
0answers
33 views

Fixing learning rate for gradient descent single variable

I need guaranteed convergence to local minimum given initial value in $(0,6)$. The function is $f(x) = 30-61 x+41 x^2-11 x^3+x^4$. I have $x(i+1) = x_i - \eta (4x_i^3-33x_i^2+82x_i - 61) $. What ...
0
votes
1answer
30 views

How to express $\sum_{i=0}^{n}a_i^m$ in terms of $\sum_{i=0}^{n}a_i$ and $\sum_{i=0}^{n}a_i^2$

Is there any way to express $\sum_{i=0}^{n}a_i^m$ by polynomial of $X=\sum_{i=0}^{n}a_i$ and $Y=\sum_{i=0}^{n}a_i^2$? For example, if $n=2$ and $m=3$, it can be expressed as $\dfrac{X}{2}(X^2+Y)$. I ...
3
votes
0answers
69 views

Conditions for polynomial $f$ such that $f(n) \in \mathbb{N}$ for enough $n \in \mathbb{N}^+$ implies $f$ has rational coefficients

This question is suggested by this one: prove: coefficients of $f(x)$ are rational numbers What are the weakest sufficient conditions and strongest necessary conditions on a set of positive integers ...
2
votes
0answers
155 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
12
votes
2answers
234 views

Proving that $\sum_{k=1}^{\infty} \frac{3408 k^2+1974 k-720}{128 k^6+480 k^5+680 k^4+450 k^3+137 k^2+15 k} = \pi$

I am trying to prove that $$\sum_{k=1}^{\infty} \frac{3408 k^2+1974 k-720}{128 k^6+480 k^5+680 k^4+450 k^3+137 k^2+15 k} = \pi$$ This is what I've tried to simplify the sum: $$\frac{3408 k^2+1974 ...
1
vote
2answers
84 views

Is there a way to fit an even function using only odd functions?

I was wondering if there is a way to make an infinite series of odd functions equal to an even function. For example, I would like to know if the next equation is valid ...
6
votes
2answers
206 views

Polynomial long division: different answers when reordering terms

When I use polynomial long division to divide   $\frac{1}{1-x}$,   I get $\;1 + x + x^2 +x^3 + x^4 + \cdots$ But when I just change the order of terms in the divisor:   ...
4
votes
1answer
151 views

Solve $a^3 + b^3 + c^3 = 6abc$

Find solutions for $a^3 + b^3 + c^3 = 6abc$ in $\mathbb{N}$, such that $gcd(a,b,c) = 1$, except for $(1,2,3)$ and its permutations. Using trial and error I found out that if $a,b,c$ are solution ...
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 ...
2
votes
0answers
185 views

Prime number finding via polynomials

I try to find approximation polynomial to estimate which number is prime or not. Addtion to this, (If It is possible) To find the closed form of coefficients of the series ($c(n)$) Euler found the ...
0
votes
1answer
71 views

What is the pattern of this sequence?

I went though this pattern and I think the results might be interesting. It was a long one but I'm only showing the first five (to make things look simpler). $$0,1,a+b,a^2 + b^2 + \frac 32ab , ...
4
votes
4answers
372 views

Finding formula for partial sum of polynomial terms?

For example, we know that the following is true (and can be easily derived): $\sum\limits_{x=1}^{n}x = \frac{1}{2}n(n+1)$ But, what if we wanted to find the sum of a series like this: ...
1
vote
1answer
300 views

Summation with factorial terms (involving Laguerre polynomials)

As part of an exercise including gamma functions and Laguerre polynomials, I need to show that for a Laguerre polynomial $L_n(x)$, $$\int\limits_0^\infty L_n(x)x^ke^{-x}dx = 0 \textrm{ with } n \in ...
2
votes
2answers
120 views

Prove the following:

$$\sum_{k=1}^{\infty }\frac{1}{(2k-1)^{2}}=\frac{\pi ^{2}}{8}$$ I don't really know how to prove this, will assuming that $$cos(x)=\sum_{k=0}^{\infty }(-1)^{k}\frac{x^{2k}}{(2k)!}$$ help?
1
vote
1answer
33 views

A question about digital sum of polynomials over $\mathbb Z^+$

Given a polynomial with positive integer coefficients , let $a_n$ be the sum of the digits in the decimal representation of $f(n)$ , $nāˆˆ\mathbb Z^+$ , then is it true that there is a number which ...
1
vote
1answer
84 views

what if geometric sequence + geometric sequence

I wrote a program that basicly can find the formula of the sequence that created with any-degree equation. For example if you give my program that sequence: [1926, 2811, 833240, 28778265, 398155842, ...
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: $$-\left(\frac{1}{x}\right)-\left(\frac{1}{x}\right)^2-\left(\frac{1}{x}\right)^3-\left(\frac{1}{x}\right)^4+...$$ ...
4
votes
1answer
217 views

How to combine/manipulate two summations into one summation in general?

I always struggle to understand what I can and can't do with sums. In fact, even when convergence isn't an issue, I still get confused. What can I do about this problem? For instance, I am currently ...
0
votes
1answer
56 views

Geometric sequence for $a$ and $b$

$a$, $a^{\log_{10}a}$, $b^{\log_{10}b}$, $(ab)^{\log_{10}(ab)}$ are successive terms of a geometric sequence. Find the values of $a$ and $b$. What I've tried so far: ...
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 ...
-1
votes
2answers
83 views

How prove the statement that a infinity derivable function has a Taylor expansion. [closed]

What is the proof of the following statement (Taylor statement): An infinity derivable function (like $\sin x$ or $\cos x$) has a infinite polynomial series that approaches it.
17
votes
1answer
253 views

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes. I've been thinking about this question, but ...
1
vote
3answers
110 views

A sequence polynomial $P_n(x)$

Given the polynomial sequence $(P_n(x))$ satisfying $$P_0(x)=P_1(x)=1$$ $$P_{n+2}(x)=P_{n+1}(x)+xP_n(x)$$ Find $P_n(x)$ I know $P_n(x)=\sum_{k\ge 0} {n-k\choose k}x^k$ but don't know how to solve ...
2
votes
1answer
145 views

Convergence test with polynomials

Let $P, Q \colon \mathbb{R} \rightarrow \mathbb{R}$ are polynomials, and $Q(n) \neq 0$ for $n\in\mathbb{N}$ Suppose, that $\deg (P) < \deg (Q)$. Prove, that infinite sum ...
3
votes
1answer
66 views

The series $\sum\limits_{n=0}^\infty {a_{n}}(x-c)^n $ is a polynomial.

The series $\sum\limits_{n=0}^\infty {a_{n}}(x-c)^n $ is a polynomial in $x$. (This is from this question.) For $c=0$ this clearly means that, for some $n>k, \space a_n=0$, almost by definition. ...
1
vote
1answer
148 views

Sequences with induction and proving. Polynomial and rational functions

$1.$We define a sequence of rational number {$a_n$} by putting $$a_1 =3,\;\text{ and}\;\; a_{n+1} = 4 - \frac{2}{a_n} \text{ for all}\; n \in \mathbb{R}.\;\text{ Put}\;\; \alpha = 2 + \sqrt{2}.$$ ...
8
votes
1answer
194 views

Do roots of a polynomial with coefficients from a Collatz sequence all fall in a disk of radius 1.5?

Consider a modified version of Collatz sequence: $C(n)=\left\{ \begin{array}{ll} \frac{3n+1}{2} & n\ \mathrm{odd} \\ \frac{n}{2}& n\ \mathrm{even}\end{array} \right.$ Let $F_n$ be the ...
3
votes
0answers
209 views

Closed-form expression for sum of Vandermonde matrix elements

Given the Vandermonde matrix: $$\begin{pmatrix}1^0 & 1^1 & 1^2 & ... & 1^n \\ 2^0 & 2^1 & 2^2 & ... & 2^n \\ \vdots & \vdots & \vdots & \ddots & ...
1
vote
1answer
77 views

Polynomial Formula like Infinite Sum with non-natural index

By polynomial formula $$(\sum_{i\in m} x_i)^n=\sum_{\substack{j_i \in \mathbb{Z}^+ \\ \sum j_i=n}}\left(\begin{array}{c} n\\ j_{0},\ldots , j_{m-1} \end{array}\right)\prod_{i \in m} x_i^{j_i}$$ where ...
2
votes
1answer
66 views

construct $\{a_n\}$, for which exists $\{n_k\}$ and $\sum_{n=1}^{n_k}a_nx^n$ converges uniformly to $f$, where $f\in C[0,1]$, $f(0)=0$

Construct the sequence $\{a_n\}\subset\mathbb R$ such, that for every $f\in C[0,1]$ with $f(0)=0$, there exists a sequence $\{n_k\}$ for which \begin{equation} ...