Life Contingencies School Level Maths Proof 8

Starting with the three equations:

1. 1+i=v^(-1)

2. (1+i(m)/m)^m=v^(-1) and

3. (1-d(m)/m)^m=v, show that lim v→1 (i-i(m))/(i(m)*d(m))=(m-1)/(2*m).

(i-i(m))/(i(m)*d(m)) is often called beta(m) for short.

Method

Re-arranging equation 1. gives i=v^(-1)-1, re-arranging equation 2. gives i(m)=m*(v^(-1/m)-1) and re-arranging equation 3. gives d(m)=m*(1-v^(1/m)), meaning that lim v→1 (i-i(m))/(i(m)*d(m))=lim v→1 (v^(-1)-1-m*(v^(-1/m)-1))/(m*(v^(-1/m)-1)*m*(1-v^(1/m))), which, after using an Online Limits Calculator, is found to equal (m-1)/(2*m).

Note that lim m→infinity (m-1)/(2*m), after using an Online Limits Calculator, is found to equal 1/2, and this result can, of course, also be arrived at by performing lim m→infinity, first, and then performing lim v→1, thus; lim m→infinity beta(m)=lim m→infinity (v^(-1)-1-m*(v^(-1/m)-1))/(m*(v^(-1/m)-1)*m*(1-v^(1/m))), which, after using an Online Limits Calculator, is found to equal (v^(-1)-1+ln(v))/(-ln(v))^2, and lim v→1 (v^(-1)-1+ln(v))/(-ln(v))^2, after using an Online Limits Calculator, is found to equal 1/2