please i need if i compare the origin with this function ,what we did for both side (steps)

for example the origin is 1/1-z =sum 0-infinity of z^n |z|<1

what we do for both side to get the given function ,then what the final tayolr series

f(z) = 1/(1-z) f(o) =1

f’(z) =-1/(1-z)^{2}(-1) f’(0) =1

f”(z) = (1)(-2)/(1-z)^{3}(-1) f’’(0) =2!

f ^{n} (z) = (1)(2)…(n) /(1-z)^{n+1} f^{n}(0) = n!

f(z) = f(0) +z f’(0)/1!+z^{2}f’’(0)/2!+ …z^{n }f^{n}(0)/n!+..

= 1+z+z^{2} 2!/2!+z^{3}3!/3!+…z^{n} n!/n!+…

= 1+z+z^{2}+z^{3}+………z^{n}+…

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