算法复杂度分析

算数级数：与末项平方同阶

T(n) = 1+2+…+n = n(n+1)/2 = O(n²)

幂方级数：比幂次高出一阶

T₂(n) = 1² + 2² + 3² + … + n² = n(n+1)(2n+1)/6 = O(n³)

T₃(n) = 1³ + 2³ + 3³ + … + n³ = n²(n+1)²/4 = O(n⁴)

T₄(n) = 1⁴ + 2⁴ + 3⁴ + … + n⁴ = n(n+1)(2n+1)(3n² + 3n -1)/30 = O(n⁵)

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几何级数(a > 1):与末项同阶

T_a(n) = a⁰ + a¹ + … + aⁿ = (aⁿ⁺¹ -1)/(a - 1) = O(aⁿ)

1 + 2 + 4 + … + 2ⁿ = 2ⁿ⁺¹ - 1 = O(2ⁿ⁺¹) = O(2ⁿ)

收敛级数

1/1/2 + 1/2/3 + 1/3/4 + … + 1/(n-1)/n = 1 - 1/n = O(1)

1 + 1/2² + … + 1/n² < 1 + 1/2² + … = π²/6 = O(1)

1/3 + 1/7 + 1/8 + 1/15 + 1/24 + 1/26 + 1/31 + 1/35 + … = 1 = O(1)

可能未必收敛，但是长度有限

调和级数 : h(n) = 1 + 1/2 + 1/3 + … + 1/n = O(log(n))

对数级数 : log(1) + log(2) + log(3) + … + log(n) = log(n!) = O(nlog(n))