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The practical use of the formula has been demonstrated for gambling,[2][3] and the same idea was used to explain diversification in investment management. Some corrections have been published. lim N → + ∞ K N = p {\displaystyle \lim _{N\to +\infty }{\frac {K}{N}}=p} The fraction of the bettor's funds to bet on k k -th horse is f k f_{k} . The formula for the optimal fraction f k o f_{k}^{o} may be interpreted as the excess of the expected revenue rate of k k -th horse over the reserve rate divided by the revenue after deduction of the track take when k k -th horse wins or as the excess of the probability of k k -th horse winning over the reserve rate divided by revenue after deduction of the track take when k k -th horse wins. ∑ i p i < 1 , {\displaystyle \sum _{i}{p_{i}}<1,} ∑ i β i < 1 \sum _{i}{\beta _{i}}<1 {\displaystyle \mathbb {E} \left[\ln \left((1+r)+\sum \limits _{k=1}^{n}u_{k}(r_{k}-r)\right)\right]. Expanding this with a Taylor series around u 0 → = ( 0 , . designer bags cyber monday