[SOLVED] Complex Analysis: Limits!!!

• Feb 15th 2009, 04:35 AM
syster
[SOLVED] Complex Analysis: Limits!!!
What is the limit and n tends to infintity of
z+z^2+z^3+...+z^n
note z belongs to the complex plane and is defined on the disc with centre 0 and radius 1.

I think it should be 1/(1-z) but I cant see why.

Thanks
• Feb 15th 2009, 06:42 AM
andreas
Quote:

Originally Posted by syster
What is the limit and n tends to infintity of
z+z^2+z^3+...+z^n
note z belongs to the complex plane and is defined on the disc with centre 0 and radius 1.

I think it should be 1/(1-z) but I cant see why.

Thanks

use this fact: the sum is equal to ${z(1-z^{n+1})}/(1-z)$ where n-> $\infty$. The expression can be expanded to $z/(1-z)+z^{n+2}/(1-z)$ taking the limit n-> $\infty$ second term vanishes. That's it!