sin^3(n)/n级数不绝对收敛,求证

Question

sin^3(n)/n 级数不绝对收敛 ,求证

Answer 1

|sin^3n/n|>=sin^4n/n=(1-cos2n)^2/4n=(2-4cos2n+1+cos4n)/8n=3/8n-cos2n/2n+cos4n/8n,用Dirichlet判别法可以判别级数cos2n/2n和cos4n/8n是收敛的,而级数3/8n是发散的,故级数|sin^3n/n|是发散的.