Let fz be an entire function such that fn1z equivalent 0 Pro

Let f(z) be an entire function such that fn+1(z) equivalent 0, Prove that f is a polynomial of degree at most n?

Solution

We can prove the above formula using mathematical induction.
So lets us take multiple cases.
For f(x) = x, f(x)\'=1, f(x)\'\'=0. so for f(x)^2 the polynomial power is 1,
for f(x)=x^2, f(x)\'=2x, f(x)\'\'=2, f(x)\'\'\'=0, so for f(x)\'\'\' the polynomial is 2.

So we can deduce that for a polynomial with n power it should be differentiated n+1 times to get fn+1(z)=0,
So it could be stated otherwise if fn+1(z)=0 then f(z) has a power of atmost x^n.