Toronto Math Forum

MAT334-2018F => MAT334--Lectures & Home Assignments => Topic started by: Nikki Mai on November 19, 2018, 09:37:13 PM

Title: 2.2 Q11
Post by: Nikki Mai on November 19, 2018, 09:37:13 PM

Can anyone help me with 2.2 question11?
I am not sure how to solve with the hint.

Title: Re: 2.2 Q11
Post by: Ye Jin on November 19, 2018, 09:59:15 PM

    $\frac{1}{4-z}=\frac{1}{4}\frac{1}{1-\frac{z}{4}}$
    $\frac{1}{4-z}=\frac{1}{4}\sum_{n=0}^{\infty}(\frac{z}{4})^n$
   
   Take derivatives on both sides, then
    $\frac{1}{(4-z)^2}=\sum_{n=1}^{\infty}\frac{1}{4^{n+1}}nz^{n-1}$  (Here, use the hint)
   
    $\frac {z^2}{(4-z)^2}=\sum_{n=1}^{\infty}\frac{1}{4^{n+1}}nz^{n+1}$
             
              $=\sum_{n=1}^{\infty}n(\frac{z}{4})^{n+1}$


OK Fiexd it.

Title: Re: 2.2 Q11
Post by: Victor Ivrii on November 20, 2018, 06:24:15 AM

$\frac{1}{4^{n+1}}$ cannot be outside of the sum