what happens to this equation after change of variable?

Question

I have the following equation $f$:

$$ f(\frac{1}{z}) = 1 + \frac{1}{z} + \frac{1}{z(z+1)} + \frac{1}{z(z+1)(z+2)} + ... $$

If I make a change of variable $\frac{1}{z}=k$, does this mean that the equation becomes ?

$$ f(k) = 1 + k + k(k+1) + k(k+1)(k+2) + ... $$

If not, what should the form for $f(k)$ be?

Answer 1

$f(k)=k+\frac 1 {\frac 1 k (\frac 1 k +1)}+\frac 1 {\frac 1 k (\frac 1 k +1)(\frac 1 k +2))}+...=k+\frac {k^{2}} {k+1}+\frac {k^{3}} {(k+1)(2k+1)}+...$.

The general term is $\frac {k^{n+1}} {(k+1)(2k+1)...(nk+1)}$

Answer 2

Hint:

As you can check by yourself,

$$\frac1{z(z+1)(z+2)}=\frac1{\dfrac1k\left(\dfrac1k+1\right)\left(\dfrac1k+2\right)}=\frac{k^3}{(k+1)(2k+1)}.$$