Question

The generating function of the sequence {1,2,3...n..} is (1-z)².

Answer 1

No.

We know that

$\sum_{n=0}^{\infin} {z^n} =\frac 1 {1-z}$

Find derivative of it:

$\sum_{n=1}^{\infin} {n*z^{n-1}} =\frac 1 {(1-z)^2}$

Change n to n+1

$\sum_{n=0}^{\infin} {(n+1)*z^{n}} =\frac 1 {(1-z)^2}$

Hence, the generating function of the sequence {1,2,3...n..} is $\frac 1 {(1-z)^2}$