# Math Help - complex nos analysis

1. ## complex nos analysis

can someone help me with this question pls think i undersdtand trhe concepts but cant quite get the answer thanks

Edgar

Let an be a sequence of complex numbers such that sigma|an| converges and
sigma (an/k^n) = 0 for all k ∈ N. Show that an = 0 for all n.

2. Originally Posted by edgar davids
can someone help me with this question pls think i undersdtand trhe concepts but cant quite get the answer thanks

Edgar

Let an be a sequence of complex numbers such that sigma|an| converges and
sigma (an/k^n) = 0 for all k ∈ N. Show that an = 0 for all n.
Put $g(k)=\sum a_n/k^n$, then if $z=1/k$:

$
f(z)=g(1/z)=\sum a_n z^n\equiv 0
$

is absolutly convergent inside the unit disk so f is analytic there, and:

$
D^{m}f(0)=m!\ a_m
$

but:

$
D^{m}f(0)=0
$

so $a_m=0$ for every $m=0, 1, ..$

RonL