# complex nos analysis

• Nov 22nd 2006, 03:53 AM
edgar davids
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.
• Nov 22nd 2006, 08:18 AM
CaptainBlack
Quote:

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 $\displaystyle g(k)=\sum a_n/k^n$, then if $\displaystyle z=1/k$:

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

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

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

but:

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

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

RonL