Question

If [infinity] cn8n n = 0 is convergent, can we conclude that each of the following series is convergent? (a) [infinity] cn(−3)n n = 0 When compared to the original series, [infinity] cnxn n = 0 , we see that x = here. Since the original for that particular value of x, we know that this . (b) [infinity] cn(−8)n n = 0 When compared to the original series, [infinity] cnxn n = 0 , we see that x = here. Since the original for that particular value of x, we know that this

Answer 1

Answer:

a) we know that this is convergent.

b) we know that this might not converge.

Step-by-step explanation:

Given the $\sum^\infty_{n=0}C_n8^n$ is convergent

Therefore,

(a)  $\sum^\infty_{n=0}C_n(-3)^n$ The power series $\sum C_nx^n$ has radius of convergence at least as big as 8. So we definitely know it converges for all x satisfying -8<x≤8. In particular for x = -3

∴ $\sum^\infty_{n=0}C_n(-3)^n$  is convergent.

(b) $\sum^\infty_{n=0}C_n(-8)^n$ -8 could be right on the edge of the interval of convergence, and so might not converge