# Questions tagged [roots]

Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots" and such, consider using the (radicals) and the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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### Searching simple identities between power sums of roots?

The Newton-Girard identities relating Vieta coefficients $e_k$ with power sums $P_k$ of roots of a polynomial, namely,$$ke_k = \sum_{i = 1}^k (-1)^{i - 1}e_{k-i}P_k$$ {with $e_1$ = $P_1$, and $e_k$ = ...
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### Prove that pole of infinite series “explodes” to $+\infty$

The answer to the question at Proving that pole of infinite series "goes to" $+\infty$ or $-\infty$ wonders me how to do it if I cant pull out the numerator from the summation because it depends on $k$...
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### Proving that pole of infinite series “goes to” $+\infty$ or $-\infty$

Lets define the infinite series: $$f(x) = \sum\limits_{k=1}^{\infty} \frac{x^s}{(1-x^k)^2}, \quad s \in \mathbb{N}_0$$ For me, it is clearly to see that it has two poles: at $x = -1$ and at $x = 1$. ...
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### Absolute summability of summation of integer exponents of polynomial roots (BIBO stability)

Let $\lambda _{i} = g_{i}\angle\theta_{i}$ be the roots of a M degree polynomial $S= \sum _{n=0}^{\infty} \left ( \left | \sum _{r=1}^{M} \left ( K_{r} \lambda _{r}^{n}\right )\right | \right )$ , ...
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### Prove solution to $\sum\limits_{k=1}^{\infty}\frac{x^k}{x^{2k}-2x^k+1} = 0$.

Numerically I found a solution to the following equation at about $x = -0.4112$. $$\sum\limits_{k=1}^{\infty}\frac{x^k}{x^{2k}-2x^k+1} = 0 \quad x \in \mathbb{R}, -1 < x \leq 0$$ Now, I want to ...
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### Counting the roots of $f(z)=z^4+z^3-2z^2+2z+4$

I have a polynomial $f(z)=z^4+z^3-2z^2+2z+4$, and I want to find the number of roots in the first quadrant. I'm trying to use the argument principle (or Rouche), and I could try to make my contour the ...
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### Roots of $x^n+x+1$

I am trying to prove that $x^n+x+1$ does not have any real roots if n is even, and that it has just one root if n is odd. My attempt: When n is odd, I can use Bolzano's theorem to prove that there is ...
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### Root locus method, deriving the position of the asymptote centroid

In the Root Locus Method, the linear asymptotes are centered at a point on the real axis given by $$\sigma_A=\frac{\sum_{j=1}^n(-p_j)-\sum_{i=1}^M(-z_i)}{n-M},$$ where $p_j$ are the $j$th open-loop ...
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### How to derive the condition? (I am not sure it is correct.)

Given a polynomial with real coefficients $\alpha x^2 + \beta x + a^2 + b^2 + c^2 - ab- bc - ca$ has imaginary roots, how do we prove $2(\alpha - \beta) + ((a - b)^2 + (b - c)^2 + (c - a)^2) > 0$...
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Let $\theta$ be an integer bigger or equal to three, let $\lambda^M \ge 0$ . In addition to this let $\left\{ \zeta_j \right\}_{j=1}^\theta$ be roots of a following polynomial equation $x\cdot (x-1) \... 3answers 63 views ### Is there some theorem related to the triangle inequality that says$\sqrt{x+y}\leq \sqrt{x} + \sqrt{y}$? I know that the triangle inequality says$|a+b| \leq |a| + |b|$. But what about square roots? 3answers 75 views ### How do I solve the equation$ 2x^3+6x^2-12x-4=0$[closed] I tried to group the terms like this $$(2x ^ 3 + 6x ^ 2) - (12x + 4) = 0$$, but I couldn't give a common factor. 0answers 35 views ### Please help me find the zeros of the following function Let$g(z)=e^{z^2}-\frac{z}{e^{1+2z}-1}$. Clearly,$g\not\equiv 0$since the function$\frac{z}{e^{1+2z}-1}$is meromorphic and also$e^{z^2}$is entire. But, what can be said about the zeros of$g(z)$?... 0answers 14 views ### Find Roots of$f(\theta) = \theta^{\delta-1}(1-\theta)^{\beta-1}-\tau$Let $$f(\theta) = \theta^{\delta-1}(1-\theta)^{\beta-1}-\tau$$ The first term in$\theta$proportional to a beta density. I am interested in finding the roots of$f(\theta)$. According this post, ... 2answers 75 views ### Solution of$(1-x)^n = x$: Rate of convergence of$x \to 0$as$n \to \infty$? The equation$(1-x)^n = x$has a solution in$x' \in (0,1)$and indeed the solution$x' \to 0$as$n \to \infty$. (Consider$f(x) = (1-x)^n$noting that$f(0) = 1$and$f(1) = 0$. As$n$increases,$...
I'm looking for a way to show that an equation like $$15(1+x+x^{2})(1+y+y^{2})=16(xy)^{2}-1$$ has no solutions with x and y odd positive integers. The 16 and 15 can be changed but I'm trying to ...