Question

Suppose R>0. Which of the following statements might be true about a power series centered about the point x=a

a) The series converges for all x>a

b) The series converges on the interval (a,R)

c)The series converges on the interval [a,R)

d) The series converges only at x=a+R

e) All of the above

f) None of the above

Answer 1

The power series centered about the point x = a satisfies exactly one of the below properties.

1- The series converges for all real numbers.

2- The series converges at x = a and diverges for all other real numbers.

3- There exists a real number R > 0 such that for |x - a| < R, the series converges and for |x - a| > R, the series diverges. The series may converge or diverge at x = a - R and x = a + R.

Hence, for R > 0, the only statement that might be true about a power series centered about the point x = a is

f) None of the above