Question

工科数学分析下册（丁晓庆）,习题21.3,15设{an}是严格单调减收敛于0的数列,记An=(a1+a2+……an)/n

Answer 1

设 Sn=a1+a2+……an.则 S_{n+1}=S_n+a_{n+1}.由于 a_n 严格单调递减,故 S_n>n a_{n+1} (因为 a_1>a_2>...>a_n>a_{n+1}.

A_{n+1}-A_n=S_{n+1}/(n+1)- S_n/n= (S_n+a_{n+1})/(n+1)- S_n/n = a_{n+1}/(n+1)- S_n/(n(n+1))