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The “multiplier of the variance ” function grows very slowly. To understand how very slowly, calculate a table with n = 10^{j} and for j = 10, 20, 30,..., 100. How large must n be for 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.)
j | n = 10^{j} | |
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 to exceed 10, n must be larger than e^{e50} e^{5.18×1021}. This is a number which would have about 2.24 × 10^{21} decimal digits.
For to exceed 20, n must be larger than e^{e50} e^{7.23×1087}. This is a number which would have about 3.13 × 10^{87} decimal digits.
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