Why are these estimates to the German tank problem different?

Suppose that I observe $k=4$ tanks with serial numbers $2,6,7,14$.
What is the best estimate for the total number of tanks $n$?

I assume the observations are drawn from a discrete uniform distribution with the interval $[1,n]$. I know that for a $[0,1]$ interval the expected maximum draw $m$ for $k$ draws is $1 – (1/(1+k))$. So I estimate $\frac {k}{k+1}$$(n-1)≈$ $m$, rearranged so $n≈$ $\frac {k+1
}{k}$$m+1$.

But the frequentist estimate from Wikipedia is defined as:

$n ≈ m-1 + $$\frac {m}{k}$

I suspect there is some flaw in the way I have extrapolated from one interval to another, but I would welcome an explanation of why I have gone wrong!

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