Originally Posted by

**NoMoreExams**
This is easy to figure out

Complete: e_{x:n} = integral(t_p_x dt from 0 to n) = integral(1 - t/(w-x) dt from 0 to n) = n - n^2/(2(w-x)) = n[1 - n/[2(w-x)]]

Curtate (not Kurate): e_{x:n} = sum(t_p_x from t=1 to n - 1) = sum(1 - k/(w-x) from 1 to n-1) = (n-1) - n(n-1)/[2(w-x)] = (n-1)[1 - n/[2(w-x)]] = n[1 - n/[2(w-x)]] - [1 - n/[2(w-x)]]

So as you can see Curtate = Complete - [1 - n/[2(w-x)]]

Obviously if the sum goes to "infinity" i.e. in this case to w - x, we let n = w-x and we get:

Curtate = Complete- [1 - (w-x)/[2(w-x)]] = Complete - [1 - 1/2] = Complete - 1/2

Hopefully my algebra is right.