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

Answer:

A. FALSE

B. TRUE

C. FALSE

D. NOT POSSIBLE TO DETERMINE

Step-by-step 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.