设数列{an}满足不等式0

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设数列{an}满足不等式0

考察当m=2n时的m*a(m)
1.
∵0≤ak≤100an,(n≤k≤2n,n=1,2,...)
∴0≤a(2n)≤100a(n+p),(p=1,2,...,n)
其中n+p≤2n ≤2n,n=1,2,...,
2.
m*a(m)
=(2n)*a(2n)
=2*a(2n) +2*a(2n) +2*a(2n) +…2*a(2n) ,(有n个2*a(2n) )
≤200[a(n+1) +a(n+2) +a(n+3) +…+a(2n)]
3.因为无穷级数a1+a2+…+an+…收敛,
所以根据Cauchy收敛准则,当n→∞时,
[a(n+1) +a(n+2) +a(n+3) +…+a(2n)] → 0
从而(2n)*a(2n) → 0
4.同样可以证明:
当m=2n-1时,m*a(m) → 0,(当n→∞时);
从而n*a(n) → 0,(当n→∞时).

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