※ 引述《buttermilk (脫脂牛奶)》之銘言：
: 1.設 p 與 q 為大於1的正整數，若 p>q且x>0，證明：(x^p-1)/p ≧(x^q-1)/q。
define f_n(x) = (1+x+x^2+...+x^{n-1})/n
Claim : f_n(x) is increasing when x>1
Proof:
n*(n+1)*[f_{n+1}(x) - f_n(x)]
= n(1+x+x^2+...+x^{n-1}+x^n)-(n+1)(1+x+x^2+...+x^{n-1})
= n x^n - (1+x+x^2+...+x^{n-1}) > 0
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Similarly, f_n(x) is decreasing when x<1
and (x^n - 1)/n = (x-1)*f_n(x)
done
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