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

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

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

f(z) = e^{z } f(o) =1

f’(z) =e^{z } f’(0) =-1

f”(z) = e^{z } f’’(0) =1

f ^{n} (z) = e^{z} f^{n}(0) = 1

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

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

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