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

In summary, the Taylor series of a function, around a certain point, converges over an interval symmetric about that point until a point where the function is not "analytic". For the given function (a) sqrtX^2-x-2 about x=1/3, the series will converge for all values of x between 2 and 4. For the function (b) sin(1-x^2) about x=0, the series will converge for all values of x. It is also important to note that every power series, of the form \sum p_n(x- a)^n, converges for at least x= a.
  • #1
elliegurl297
2
0
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
 
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  • #2
elliegurl297 said:
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. \(\displaystyle x^2- x- 2= (x- 2)(x+ 1)\) 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 [itex]\sqrt{x^2- x- 2}[/itex], 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 [itex]\sum p_n(x- a)^n[/itex] converges for at least x= a!

and for b) I am not sure

any help would be appreciated
thankyou
ellie
 

1. What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of its derivatives evaluated at a specific point. It is used to approximate a function within a certain interval by adding together terms of increasing degrees of the function's derivatives.

2. How is a Taylor series different from a Maclaurin series?

A Taylor series is a more general form of a Maclaurin series, which is a special case of a Taylor series where the point of evaluation is at x=0. This means that a Maclaurin series only includes non-negative powers of x, while a Taylor series can include both positive and negative powers of x.

3. What is the convergence of a Taylor series?

The convergence of a Taylor series refers to how well the series approximates the original function. It is determined by the radius of convergence, which is the distance from the point of evaluation at which the series will converge.

4. How do you find the Taylor series for a given function?

The Taylor series for a function can be found by taking the derivative of the function and evaluating it at the point of interest, then dividing by the factorial of the derivative's order. This process is repeated for each subsequent derivative, with the power of x increasing by one each time. The resulting terms are then added together to form the Taylor series.

5. How can you determine if a Taylor series accurately represents a function?

The accuracy of a Taylor series can be determined by comparing it to the original function or by using the remainder term of the series. The remainder term calculates the difference between the original function and the Taylor series, and can be used to determine the maximum error of the approximation.

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