power series, uknown substitution of $-1$

Question

This time I type my problem into LaTeX, but to show the problem I'll also include the appropriate snippet:

I do not understand in the book by Ralston: A first course in numerical analysis $(4.10-28)$ why

$$\Sigma_{j=0}^m |c_{mj}|=t_m(-1):$$

I would understand it if $c_{mj}$ alternate their signs,starting with positive for $j=0$, but I do not know why this is the case.

score 2 · Accepted Answer · answered Jan 01 '23 at 17:48

2

So your question is, if $x_1,...,x_n$ are positive real numbers and $$ p(z)=\sum c_kz^k=(x_1-z)(x_2-z)...(x_n-z), $$ if then the coefficient sequence is alternating. This is quite easy to see, as in the expansion of $$ p(-z)=\sum c_k(-1)^kz^k=(x_1+z)(x_2+z)...(x_n+z) $$ all coefficients are positive.

answered Jan 01 '23 at 17:48

Lutz Lehmann

131,652

Right! Have you thought about which book may I read in succession of Ralston, which is as good as that but more modern ? One or two hints will do. Thank you. – user122424 Jan 01 '23 at 21:16
Dear Lutz, I've started a bounty. Please, consider replying. – user122424 Jan 06 '23 at 19:06

power series, uknown substitution of $-1$

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