Actuarial Outpost Textbook exercise 2.1g
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#1
08-13-2018, 07:33 PM
 gaudettj Member SOA Join Date: Nov 2013 Location: Canada Studying for FAP / C Favorite beer: All of the Above Posts: 246
Textbook exercise 2.1g

Maybe I'm missing something here, but the solution manual calculates ∑ (1 - x/55)^(1/5) from 1 to 54 as 45.18 without any steps. My handy dandy Wolfram Alpha gives the correct answer though no steps... Is this something that should easy to compute by hand?
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#2
08-13-2018, 10:17 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,926

There's no analytical formula for the sum. It is monotonic decreasing so the value would be greater than the integral from 1 to 55 and less than the integral from 0 to 54. You would probably get a good approximation by averaging the two integrals.
#3
08-15-2018, 03:23 PM
 gaudettj Member SOA Join Date: Nov 2013 Location: Canada Studying for FAP / C Favorite beer: All of the Above Posts: 246

Quote:
 Originally Posted by Academic Actuary There's no analytical formula for the sum. It is monotonic decreasing so the value would be greater than the integral from 1 to 55 and less than the integral from 0 to 54. You would probably get a good approximation by averaging the two integrals.
Thanks for the reply. Definitely did not think of using an approximation like this... Evidence of my rusty math skills lol
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#4
08-15-2018, 04:55 PM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,719

Quote:
 Originally Posted by gaudettj Thanks for the reply. Definitely did not think of using an approximation like this... Evidence of my rusty math skills lol
Computing the sum with Excel gives 45.1767514..., which rounds to the textbook's 45.18. The suggested approximation gives 45.1473... , which is only off by 0.065%---not bad!
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