右
ζ(2) = 1 + 1/4 + 1/9 + Σ[k=4,∞] 1/kk
　< 49/36 + Σ[k=4,∞] 1/(kk-1/4)
　= 49/36 + Σ[k=4,∞] {1/(k-1/2) - 1/(k+1/2)}
　= 5/6 + 19/36 + 2/7
　= (5 + 205/42) /6,

∴　6ζ(2) < 5 + 205/42 < 5 + 44/9 < 5 + 2√6 = (√2 + √3)^2,

　6 - (22/9)^2 = 2/81 > 0　より　√6 > 22/9,

左
ζ(2) = Σ[k=1,∞] 1/kk
　= 2 - Σ[k=1,∞] {2/(2k-1) -2/(2k+1) -1/kk}
　= 2 - Σ[k=1,∞] {4/(4kk-1) - 1/kk}
　= 2 - Σ[k=1,∞] 1/{(4kk-1)kk}
　= 2 - 1/3 - 1/60 - 1/315 - Σ[k=4,∞] 1/{(4kk-1)kk}
　> 2 - 89/252 - (1/63)Σ[k=4,∞] 1/kk
　= 2 - 89/252 - (1/63){ζ(2) - 49/36},

∴　6ζ(2) > 10 - 7/48 = 9.854167