Question

stuck on this one:
show that \[\Large \sum_{n=0}^{\infty}\frac{ z^n }{ 2^{n+1} }\] and \[\Large \sum_{n=0}^{\infty}\frac{ (z-i)^n }{ (2-i)^{n+1}}\] are analytic continuations of each other

Answer 1

ok i can show its analytic , but what does continuations of each other
of each other mean?

Answer 2

same here
i'm doing this kinda prob. for the first time
wiki says something about it-- http://en.wikipedia.org/wiki/Analytic_continuation

Answer 3

i think we have to show that those power series represents a same function @ikram002p

Answer 4

for this function
\[\Large \sum_{n=0}^{\infty}\frac{ z^n }{ 2^{n+1} }\]
i used ratio test and found out that this series converges for \( |z|<2\)

which is a geometric series with first term 1/2 and common ratio Z/2
so it will be=----
\[\frac{ 1/2 }{ 1-(z/2) }=1/(2-z)\]
it is correct? @ikram002p