QUIZ 6.4B

Theorem 6.5 If a power series anzn converges when z = z1 ( 0) then it is A.C. for all  z : | z | < | z1 |.

Proof  Since anz1n is convergent, for some M we have { 1 } for all n. We write { 2 }.

Then                        | anzn | = { 3 } < Mkn.

Now the series with terms { 4 } is a real, convergent, geometric series.
So by the Comparison Test,  anzn is convergent.

Match the above boxes 1, 2, 3, 4 with the selections
(a) Mkn (k < 1) ; (b) | anz1n |.| z / z1 |n ; (c) | anz1n | < M ; (d) | z | / | z1 | = k ( < 1 ).

My solutions:

1.   ;  2.   ;  3.   ;  4.   .   Check