So sánh :3/1^2.2^2+5/2^2.3^2+7/3^2.4^2+...=19/9^2.10^2<1

312⋅22+522⋅32+732⋅42+…+1992⋅102 và 1

Ta có:

312⋅22+522⋅32+732⋅42+…+1992⋅102=191−122+132−132−142+…+192−1102=1−1102=99100<1

Vậy 312⋅22+522⋅32+732⋅42+…+1992⋅102<1