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Consider the set of polynomials {z^n} where n ranges from 0 to infinity using even integers. All the zeros of the polynomials in the set are zero and so stable
by shunted 19y ago
Consider the set of polynomials {z^n} where n ranges from 0 to infinity using even integers. All the zeros of the polynomials in the set are zero and so stable. Consider the power series expansion of .5 Cos(z). It is a weighted sum of the polynomials of our set and the zeros of .5 Cos(z) are clearly not bounded. I use .5 Cos(z) so that the coefficient of each term of the power series expansion is a value that satisfies 0 < a < 1.
- aswanson 19y agoConsider the set of polynomials {z^n} where n ranges from 0 to infinity using even integers. All the zeros of the polynomials in the set are zero and so stable But consider the case where n=0,2. Then you have z^0+z^2, with a zero at +/- j. So that statement isn't totally true.
- shunted 19y agoI provided an example where the answer to the question posed is false. There are examples where it is true. Since a counterexample exists then the conjecture is considered false. A conjecture can be false even though there exist instances where it is true.
- pixcavator 19y agoHe probably meant that the sequence of partial sums should be stable, as a whole.