not sure whether i understand the problem, what does Sm(-1) look like? In fact, what does S_m (-n) look like? Is it (-1)^m+(-2)^m+....+(-n)^m?
He meant show that the polynomials that the power series sum (taken as polynomials defined for any real n) to have -1 and 0 as roots.not sure whether i understand the problem, what does Sm(-1) look like?
I could be misunderstanding, but from the definition of S_m (n) (in the first line of the question), does this mean that S_m (n) holds positive integer values only?He meant show that the polynomials that the power series sum (taken as polynomials defined for any real n) to have -1 and 0 as roots.
I could be misunderstanding, but from the definition of S_m (n) (in the first line of the question), does this mean that S_m (n) holds positive integer values only?
subbing in n=-1 gives 1+ sigmaS_k(-1)*{m+1)Ck=(-1+1)^m+1=0??
The first argument doesn't work, just because the sum of all of them is zero doesn't mean all of them are zero (for instance 1 + (-1) +2 + (-2) = 0 but they aren't all zero).subbing in n=-1 gives 1+ sigmaS_k(-1)*{m+1)Ck=(-1+1)^m+1=0??
the LHS is 1 if S(-1)=0 for all k?
? Not sure what you mean here? Since S_k (-1) =0 as you claimed for all integers k, doesn't this mean that sigmaS_k(-1)*{m+1)Ck equals 0? (Since all the terms are zero, their sum is zero....am i wrong here?)The first argument doesn't work, just because the sum of all of them is zero doesn't mean all of them are zero (for instance 1 + (-1) +2 + (-2) = 0 but they aren't all zero).
Try induction!
(Also yea the property "S_m(n) has roots n=0 and n=-1" is only true for m >= 1, I'll edit that in)
Breaking my Bored of Studies fast to help a fellow Boser in need.
The second part of my post was saying that it's not true that S_0(-1) = 0, so that's why I edited in the question to make it "prove for m>=1 that S_m(-1) = 0" (which is true).? Not sure what you mean here? Since S_k (-1) =0 as you claimed for all integers k, doesn't this mean that sigmaS_k(-1)*{m+1)Ck equals 0? (Since all the terms are zero, their sum is zero....am i wrong here?)
Fixed that for youBreaking my Bored of Studies fast to help a fellow Boser in need.
You should mention when the domain is restricted!
Key is the triangle inequality.
Do we have to use the triangle inequality? I thought by differentiating to find the min stationary point and subbing back into P(x) we obtain what we need?Key is the triangle inequality.
For |z| <= 1, we have
|p(z)| >= 3- |z+z^4| >= 3-|z|-|z|^4 >= 1 > 0.
and
|z^4q(1/z)| = |z^4+z^3+4| >= 4-|z|^3 - |z|^4 >=2 > 0.
So p has no roots inside the closed unit disk and q has no roots outside the open unit disk (and so certainly has no roots outside the closed unit disk).
Do we have to use the triangle inequality? I thought by differentiating to find the min stationary point and subbing back into P(x) we obtain what we need?