Convergence/Divergence quesiton

Question

Suppose the power series, ∑c_nxⁿ , 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

For Power Series, look at the ratio of successive terms:

R_n = c_n+1xⁿ⁺¹ / c_nxⁿ = c_n+1 x / c_n

The Radius of Convergence R is defined as the limit of R_n as n→ ∞.

When we say that a power series converges when x = -4, we mean that R ≥ 4.

When we say that a power series diverges when x = 7, we mean that R < 7.

We conclude that R is somewhere in the range of [ 4, 7 ).

We can now answer the questions asked:

a) False

b) True

c) False

d) Unknown