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174 MathChapter E Take N, = N N₁ N₂ - because Σ Nⱼ = N. Then we can use Stirling's approxi- mation (Equation E.7) to write In W as In W = N In N N Nⱼ In Nⱼ Nⱼ j=1 (N N₁ N₂ In(N N₁ N₂ ... + N N₁ N₂ ... Now take the partial derivative of In W with respect to N₁. We find that In W = - In N₁ + 1 1 + N₁ N₂ ... - As in Problem E-9, we set d(In W)/dN₁ = 0 to find = We get a similar result for N₂ through so N₁ = N₂ N, and all are equal to N/r. E-11. Prove that N k=0 We know that (Equation E.3) N If = y = 1, we can write this as Thus N k=0 E-12. The quantity n! as we have defined it is defined only for positive integer values of n. Consider now the function of x defined by = (1) Integrate by parts (letting и = and dv = e dt) to get = (x 1) 0 dt = (x 1) (2)