calculus help..taylor series?

Question

a) Explain why the series {1.6 - 0.8(x-1) + 0.4(x-1)^2 - 0.1(x-1)^3 + ...) is NOT the Taylor series of "f" centered at 1.

b) Explain why the series {2.8 + 0.5(x-2) + 1.5(x-2)^2 - 0.1(x-2)^3 + ...) is NOT the Taylor series of "f" centered at 2.

Answer 1

The Taylor series for F(x-a) is:

F(x-a) = F(a) + F'(a)x + (1/2!)F''(a)x^2 + ... + (1/n!)F[n]x^n + ...

So if the first series were a Taylor series for a function F centered at a = 1:

F(a) = 1.6
F'(a) = -0.8
F''(a) = (2!)(0.4)
F'''(a) = (3!)(-0.1)

or , more generally, the n-th derivative of F at a would have to be:

F[n](a) = (n!) (1.6/(2^n)) (-1)^n or
F[n](a) = F[n-1](a) (-1) (n/2)

so as n gets larger F[n] get even larger and does so faster and faster so the function F is not really well defined.

I don't see the rule for continuing the coefficients for your case b, so I can't address it.