I am not able to understand the proof of the result e_x=p_x*(1+e_x+1) can someone explain me the solution?
The first step splits the summation from k = 1 to w - x, into the value when k = 1 plus the summation from k = 2 to w - x. The second step makes the substitution in the summation of j = k - 1. This shifts the limits to j = 1 (corresponding to k = 2) and j = w - x - 1 (corresponding to k = w - x). The probability being summed is then j+1_p_x (after the substitution), and this is rewritten using the consistency condition for survival probabilities developed in Section 1.4. The third step pulls out a common factor of 1_p_x. The last step relies on spotting that the summation we now have is basically what we started with with x replaced by x +1 in the probability, so it is e_x+1.
