# Another Power Series

• Apr 10th 2010, 10:28 PM
jzellt
Another Power Series
This one seems a lot harder...

Find the power series about the origin of:

z^2/(4-z)^2
• Apr 11th 2010, 12:16 AM
mr fantastic
Quote:

Originally Posted by jzellt
This one seems a lot harder...

Find the power series about the origin of:

z^2/(4-z)^2

You only need to bother calculating the series for 1/(4 - z)^2 (why?).

Now look for the pattern in $f^{n}(0)$ where n = 0, 1, 2, 3, ....

If you need more help, please say where you are stuck.
• Apr 11th 2010, 02:39 AM
Failure
Quote:

Originally Posted by jzellt
This one seems a lot harder...

Find the power series about the origin of:

z^2/(4-z)^2

If you know the power series of $\frac{1}{(1-z)^2}$ you are essentially done. Consider:
$\frac{1}{(1-z)^2}=\frac{d}{dz}\frac{1}{1-z}=\frac{d}{dz}\sum_{n=0}^\infty z^n=\sum_{n=1}^\infty n z^{n-1}=\sum_{n=0}^\infty (n+1)z^n$

To know this is perhaps generally useful, but now let's solve your original problem:

$\frac{z^2}{(4-z)^2}=\left(\frac{z}{4}\right)^2\cdot\frac{1}{\lef t(1-\left(\frac{z}{4}\right)\right)^2}=\left(\frac{z}{ 4}\right)^2\cdot\sum_{n=0}^\infty(n+1)\left(\frac{ z}{4}\right)^n=\ldots$