Expected running time of randomized binary search

I want to calculate the expected running time of randomized binary search of the following pseudo-code, where instead of considering the midpoint as the pivot, a random point is selected:

BinarySearch(x, A, start, end)
    if(start == end)
        if(A[end] == x) 
            return end
        else
            return -1
    else
        mid = RANDOM(start, end)
        if(A[mid] == x)
            return mid
        else if(A[mid] > x)
            return BinarySearch(x, A, start, mid-1)
        else
            return BinarySearch(x, A, mid+1, end)

I looked at this previous question, which has the following:

T(n) = sum ( T(r)*Pr(search space becomes r) ) + O(1) = sum ( T(r) )/n + O(1)

How is this obtained?

sum( T(r)*Pr(search space becomes r) ) 

And in the last line of calculation, how was this obtained?

T(n) = 1 + 1/2 + 1/3 + ... + 1/(n-1) = H(n-1) < H(n) = O(log n)