(a) Formula (6) for R contains | n / n+1 |, not | n+1 / n |....

(a) Formula (6) for R contains |α_n/α_n+1|, not |α_n+1/α_n|. How could you memorize this by using a qualitative argument?

(b) What happens to R (0 < R < ∞) if you

(i) Multiply all α_n by k ≠ 0

(ii) Multiply all α_n by kⁿ ≠ 0

(iii) Replace α_n by 1/α_n? Can you think of an application of this?

(c) Which extends Theorem 2 to nonconvergent cases of α_n/α_n+1. Do you understand the principle of “mixing” by which Example 6 was obtained? Make up further examples.

(d) Does there exist a power series in powers of z that converges at z = 30 + 10i and diverges at z = 31 - 6i? Give reason.