设an=∫(0-π/4)(tanx)^ndx.求级数∑(an+a(n+2))/n的和.证明当λ>0时,∑an/n^λ收敛

题目

设an=∫(0-π/4)(tanx)^ndx.求级数∑(an+a(n+2))/n的和.证明当λ>0时,∑an/n^λ收敛

答案

a[n]+a[n+2] = ∫{0,π/4} (tan(x))^n dx+∫{0,π/4} (tan(x))^(n+2) dx= ∫{0,π/4} (tan(x))^n·(1+tan²(x)) dx= ∫{0,π/4} (tan(x))^n·(1/cos²(x)) dx= ∫{0,π/4} (tan(x))^n·(tan(x))' dx= (tan(...