Let be p(x)=a3*x^3 + a2*x^2 + a1*x + a0 so thatO((x-1)^4)=p(x) at x=1 neighborhood.

Prove: p(x)=0 for every x.

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- Mar 12th 2010, 05:40 AMAlso sprach ZarathustraTaylor problem #3
Let be p(x)=a3*x^3 + a2*x^2 + a1*x + a0 so that

**O**((x-1)^4)=p(x) at x=1 neighborhood.

Prove: p(x)=0 for every x. - Mar 12th 2010, 11:28 PMCaptainBlack
Change the variable to , then we have:

is a cubic in and that at

This means that there exists a such that:

Divide through and we have:

But for small enough we have:

So we have:

but as the left hand side goes to as goes to this is a contradiction unless .

This construction can now be repeated to show that b , which means we have proven that , and so

CB