Life Contingencies School Level Maths Proof 7

[Richard Purvey]

Life Contingencies School Level Maths Proof 7

Starting with the four equations:

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

2. 1-d=v

3. (1+i(m)/m)^m=v^(-1) and

4. (1-d(m)/m)^m=v, show that lim v→1 i*d/(i(m)*d(m))=1.

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

Method

Re-arranging equation 1. gives i=v^(-1)-1, re-arranging equation 2. gives d=1-v, re-arranging equation 3. gives i(m)=m*(v^(-1/m)-1) and re-arranging equation 4. gives d(m)=m*(1-v^(1/m)), meaning that lim v→1 i*d/(i(m)*d(m))=lim v→1 (v^(-1)-1)*(1-v)/(m*(v^(-1/m)-1)*m*(1-v^(1/m))), which, after using an Online Limits Calculator, is found to equal 1.

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