Question

Suppose that the power series, LaTeX: \sum c_n x^n∑ c n x n, converges when x = −4 and diverges when x = 7. Determine whether each statement is true, false or not possible to determine. (a) The power series converges when x = 10. (b) The power series converges when x = 3. (c) The power series diverges when x = 1. (d) The power series diverges when x = 6.

Answer 1

(a)The power series converges when x = 10: False

(b)The power series converges when x = 3: False

(c)The power series diverges when x = 1: False

(d)The power series diverges when x >= 6: False

The given information tells us that the power series converges when x = -4 and diverges when x = 7. This means that the series converges for all values of x within the interval (-4, 7).

Statement (a):

The power series converges when x = 10. This statement is false. Since the given power series diverges when x = 7, it must also diverge for any value of x within the interval (
`7, \infty` ). Therefore, the series diverges when x = 10.

Statement (b):

The power series converges when x = 3. This statement is false. Since the given power series converges when x = -4, it must also converge for any value of x within the interval (-4, 3). However, the statement does not provide any information about the interval beyond x = 3. Therefore, it is not possible to determine whether the series converges when x = 3.

Statement (c):

The power series diverges when x = 1. This statement is false. Since the given power series converges when x = -4, it must also converge for any value of x within the interval (-4, 1). Therefore, the series converges when x = 1.

Statement (d):

The power series diverges when x ≥ 6. This statement is false . Since the given power series diverges when x = 7 , it must also diverge for any value of x within the interval (7,
`\infty`). However, the statement does not provide any information about the interval between x = 6 and x = 7. Therefore, it is not possible to determine whether the series diverges when x >= 6

Complete question below:

Suppose that the power series,
`\sum c_n x^n`, converges when x = -4 and diverges when x = 7. Determine whether each statement is true, false, or not possible to determine.

(a) The power series converges when x = 10.

(b) The power series converges when x = 3.

(c) The power series diverges when x = 1.

(d) The power series diverges when x = 6.

Answer 2

A. FALSE

B. TRUE

C. FALSE

D. NOT POSSIBLE TO DETERMINE

Explanation:

(A) FALSE. since the power series ∑
`C_nx^n` has radius of convergence |-4|=4 ans 7> 4 which is beyond its radius of convergence. thus by the theorem of power series, the series diverges at 10.

(B)TRUE. since the radius of convergence of the power series ∑
`C_nx^n` must be at least |-4| = 4 and 3 lies within this radius, thus it converges at x=3

(C)FALSE the series does not diverge at x=1, since 1 is within its radius of convergence |-4| = 4

(D)NOT POSSIBLE TO DETERMINE

At x=6, it is beyond its radius of convergence but has not attain its divergence point. thus it is not possible to determine.