Finding the range of validity of a taylor series

[tomwilliam]

Homework Statement

I have to give the range of validity for a Taylor series built from an expression of the form:

(1+(a/b)x)^c

Homework Equations

The Attempt at a Solution

Obviously the validity does not extend to x=-(b/a) on the negative side, but should I then state that it is valid for:

-(b/a) < x < (b/a)

The reason being that all Standard Taylor series I've seen seem to have a symmetric interval. But I can't see why this approximation shouldn't be valid for all real numbers with the exception of -(b/a).

Any help greatly appreciated.

 

[mighty2000]

The range of validity for a Taylor series built from an expression of the form (1+(a/b)x)^c depends on the values of a, b, and c. In general, the Taylor series will be valid for all values of x within a certain interval, which may or may not be symmetric.

To determine the range of validity, you can use the ratio test or the root test to check for convergence. If the series converges, then it is valid for all values of x within the interval of convergence. However, if the series diverges, then it is not valid for any values of x.

In the specific case of (1+(a/b)x)^c, the series will converge if |(a/b)x| < 1. This means that the series will be valid for all values of x such that |x| < b/a.

In summary, the range of validity for the Taylor series will be -(b/a) < x < (b/a), as you stated, if the series converges. However, if the series diverges, then it will not be valid for any values of x.

I hope this helps. Let me know if you have any further questions.