problemset6

Problemset6

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Unformatted text preview: niqueness of 2-adic expansion to get Pn = 2n −1 � zk . k =0 3(Prob 3 on P193). Method 1. By Theorem 5, it is enough to prove ∞ � � z −z � log (1 + )e n n n=1 converges absolutely and uniformly on every compact set. Take integer M big enough such that the given compact set is bounded by M /2. Then 1+ |z | |z |2 + 2 +··· < 2 n n for n ≥ M . So ∞ ∞ �� z z� z −z � � � � � � � n ]� = �log[(1 + )e �[log(1 + ) − ]� n n n n=M n=M � � ∞ �� 1 z � 1z3 1z4 2 �[− ( ) + ( ) − ( ) + · · · ]� = � 2n � 3n 4n n=M � ∞ � � z �2 � |z | |z |2 �� ≤ + 2 +··· � � 1+ n n n n=M ∞ 1 2� 1 ≤M . 2 n2 n=M 2 This proves ∞ � � � z z log (1 + )e− n n n=1 converges absolutely and uniformly on any compact set. Method 2. By Theorem 6, it is enough to prove ∞ � �� z z � −n z −n � �e + e − 1� n n=1 converges uniformly on |z | < R for any R. This is true, since � �� ∞ �z � �� (−z )i � 1 � z 1 � −n z −n �� � − �e + e − 1� = � � i � n n i! (i − 1)! � i=2 �i−2 ∞� |z |2 � |z | 1 ≤2 n i=2 n (i − 2)! | z | 2 |z | en n2 R2 R ≤ 2 en. n ≤ 3...
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This note was uploaded on 01/22/2014 for the course MATH 18.112 taught by Professor Sigurdurhelgason during the Fall '08 term at MIT.

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