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Problem Statement

The “multiplier of the variance  V~ n-” function  V~ 2-log(log(n))- grows very slowly. To understand how very slowly, calculate a table with n = 10j and  V~ ------------ 2 log(log(n) for j = 10, 20, 30,..., 100. How large must n be for  V~ ------------- 2 log(log(n)) to exceed 10?, to exceed 20? (Remember, in mathematical work above calculus, log(x) is the natural logarithm, base e, often written ln(x) in calculus and below to distinguish it from the “common” or base-10 logarithm. Be careful, some software and technology may not properly calculate with magnitudes this large.)

Solution

j n = 10j  V~ ----------- 2 ln(ln(n))
10 1.0E+10 2.50464270
20 1.0E+20 2.76758549
30 1.0E+30 2.91040541
40 1.0E+40 3.00762760
50 1.0E+50 3.08092695
60 1.0E+60 3.13954678
70 1.0E+70 3.18826840
80 1.0E+80 3.22987897
90 1.0E+90 3.26614210
100 1.0E+100 3.29824275

For  V~ ------------- 2log(log(n)) to exceed 10, n must be larger than ee50  ~~ e5.18×1021. This is a number which would have about 2.24 × 1021 decimal digits.

For  V~ ------------- 2log(log(n)) to exceed 20, n must be larger than ee50  ~~ e7.23×1087. This is a number which would have about 3.13 × 1087 decimal digits.


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