Σ[k=1→n] a(k) = Σ[1≤4k+1≤n] [n/(4k+1)]
Σ[k=1→n] b(k) = Σ[1≤4k+3≤n] [n/(4k+3)]
よね？だから
Σ[k=1→n] a(k) - Σ[k=1→n] b(k)
= Σ[1≤4k+1≤n] [n/(4k+1)] - Σ[1≤4k+3≤n] [n/(4k+3)]
= [n/1] - [n/3] + [n/5] - [n/7] + [n/9] - … + [n/★]
= n(1 - 1/3 + 1/5 - 1/7 + 1/9 - … + 1/★)
　　　- Σ[1≤4k+1≤n] {n/(4k+1)} + Σ[1≤4k+3≤n] {n/(4k+3)}
　　　　　　　　　　　　　(∵ [α] = α - {α})
なのでもしも
(1/n) Σ[1≤4k+1≤n] {n/(4k+1)} - (1/n) Σ[1≤4k+3≤n] {n/(4k+3)}
→ 0 (n→∞)
がいえるなら求めたい極限は
1 - 1/3 + 1/5 - 1/7 + 1/9 - … → π/4
になる気がするわ

とすると
lim[n→∞] (1/n) Σ[1≤4k+1≤n] {n/(4k+1)}
lim[n→∞] (1/n) Σ[1≤4k+3≤n] {n/(4k+3)}
をどうするかが問題よね