TEAM PROJECT. Radius of Convergence. (a) Formula (6) for R contains iOn/On+li, not

Chapter 15, Problem 15.1.50

(choose chapter or problem)

Radius of Convergence.

(a) Formula (6) for R contains \(\left|a_{n} / a_{n+1}\right|\), not \(\left|a_{n+1} / a_{n}\right|\). How could you memorize this by using a qualitative argument?

(b) Change of coefficients. What happens to \(R(0<R<\infty)\) if you (i) multiply all \(a_{n}\) by \(k \neq 0\). (ii) multiply \(a_{n}\) by \(k^{n} \neq 0\), (iii) replace \(a_{n}\) by \(1 / a_{n}\) ?

(c) Example 6 extends Theorem 2 to nonconvergent cases of \(a_{n} / a_{n+1}\). Do you understand the principle of "mixing" by which Example 6 was obtained? Use this principle for making 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.)

Text Transcription:

|a_n/a_n + 1|

|a_n + 1/a_n|

R(0 < R < infty)

a_n

k neq 0

k^n neq 0

1/a_n

a_n/a_n + 1