# 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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### Existence (and uniqueness) of root of function with positive and strictly increasing derivative

Consider a differentiable function $f:\mathbb{R}\to\mathbb{R}$, and assume that: $f(a)<0$ for some $a\in\mathbb{R}$. $f'(a)\geq 0$ and $f'(x)$ is strictly increasing in $(a,+\infty)$. My question ...
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### Approximating $\sqrt[r]x$

So here's a question. What is the best way to approximate this: $$\sqrt[r]x$$ Here is a few method I found, but I am not sure which is faster: brute force. The most straight forward and easiest. Just ...
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### How to find sum of roots and product of roots with the roots given

Find the sum of roots and product of roots of a quadratic equation which has the roots -1 and 9. i often have trouble understanding the question and sometimes overthink very easy questions. Any help ...
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### Zeros of $f(x)=x^3-x$.

Zeros of the $f(x)=x^3-x$ are $x=-1, x=1$ and $x=0$. We can compute it by solving this equation : $x^3-x=0$ $x^2=1$ $x=1$ and $-1$ But when we divide both part of equation by x , we say that $x$ is ...
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### sum over $\frac{x^{k^2}}{k!}$ from $k=0$ to $\infty$

One question that I recently am the following: suppose I got the series $F(x)=\sum_{k=0}^{\infty}\frac{x^{ak^2}}{k!}$ where $a \in \mathbb{C}.$ I used Mathematica and check that this $F$ can not be ...
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### Can one show that $\sum_{k=10}^{50} a_k \cos(k \theta)$ has at least four zeroes on $[0,2\pi]$ for $a_k \in \mathbb{R}$?

This is a complex analysis puzzle that seems tricky. How can one show that for $a_k \in \mathbb{R}$, $\sum_{k=10}^{50} a_k \cos(k \theta)$ has at least four zeroes on $[0,2\pi]$? A hint is to consider ...
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### Explanation/Reference request for necessary and sufficent conditions for polynomial roots to lie inside unit circle

In the book Applied Econometric Time Series by Walter Enders (third edition, page 30) there is a discussion about the characteristic polynomial of the homogeneous part of an n-th order difference (...
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### Modifying the Ridders method with a good first estimate

I am using the ridder's method to approximate a numerical function. It has been working very well for my needs but I'm feeling that it is inefficient in that it is taking more iterations than ...
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### Is $f(x,v) = \sum_{n=0}^\infty {x^n \over \Gamma(v n +1) } > 0$ for all real $x$ and $0<v<1$?

Is $f_1(x,v) = \sum_{n=0}^\infty {x^n \over (n!)^v } > 0$ for all real $x$ and $0<v<1$ ? Lets start simple and take the case $v = 1/2$ and notice the inverse ratio of taylor coefficients is ...
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### Extensions by Adjoining elements and Extensions by quotient of a Principal Ideal

Extensions can be constructed 2 ways to get an extension with roots of a polynomial Adjoining an element to a field - i.e. $F(\sqrt 2)$ is an extension of $F$. You can also build a tower of ...