Question

设lim(n趋向∞)n（un）=0,且级数∑n(un-un-1)收敛,证明级数∑un也收敛

设lim(n趋向∞)n（un）=0,且级数∑n(un-un-1)收敛,证明级数∑un也收敛

Answer 1

简单计算一下即可，答案如图所示

Answer 2

对那个收敛级数用柯西收敛准则，再结合n(un)极限为0，可推出级数∑un满足柯西收敛准则

Answer 3

级数(n+1)(u[n+1]-u[n])收敛 那么前n项和（部分和）Sn' = 2(u[2]-u[1]) +3(u[3]-u[2])+...+(n+1)(u[n+1]-u[n]) = -2u[1]-u[2]-u[3]-...-u[n]+(n+1)u[n+1] = -u[1] -Sn + (n+1)u[n+1] 那么当n→∞时， S' = -u[1] - S + 0 其中0为nu[n]的极限. 故un收敛.