What are the expected values of x for convergence of the given Taylor series?

[elliegurl297]

Hi,
Im really stuck on my homework . The question is : For what values of x do you expect the following Taylor series to converge? Do not work out the series .
(a) sqrtX^2-x-2 about x=1/3 b) sin(1-x^2) about x=0

for a) I've put no vlues of x would the series converge. is this correct?
and for b) I am not sure

any help would be appreciated
thankyou
ellie

 

[HallsofIvy]

Hi,
Im really stuck on my homework . The question is : For what values of x do you expect the following Taylor series to converge? Do not work out the series .
(a) sqrtX^2-x-2 about x=1/3

The Taylor series of a function, around x= a, converges over an interval symmetric about a up until a point where function is not "analytic" which includes being continuous, differentiable, etc. [math]x^2- x- 2= (x- 2)(x+ 1)[/math] is negative for x between -1 and 2 and so its square root is not defined there. x= 2 is closer to 3 than -1 is so the series converges for all x between 2 and 3 and for an equal distance on the other side: the Taylor seires for \sqrt{x^2- x- 2}, about x= 3, will converge for x between 2 and 4.

b) sin(1-x^2) about x=0

This function is analytic for all x. It's Taylor's series, about any point, will converge for all x.

for a) I've put no vlues of x would the series converge. is this correct?

Every power series, of the form \sum p_n(x- a)^n converges for at least x= a!

and for b) I am not sure

any help would be appreciated
thankyou
ellie