Đặt A=1/3-2/3^2+3/3^3-4/3^4+...+99/3^99-100/3^100
3A=1-2/3+3/3^2-4/3^3+...+99/3^98-100/3^99
3A+A=1-1/3+1/3^2-1/3^3+1/3^4-...+1/3^98-1/3^99-100/3^100 <1-1/3+1/3^2+1/3^3+1/3^4-...+1/3^98-1/3^99
Đặt S=1-1/3+1/3^2-1/3^3+1/3^4-...+1/3^98-1/3^99
3S=3-1+1/3-1/3^2+1/3^3-...-1/3^98
3S+S=3-1/3^99
S=(3-1/3^99) :4
S=3/4-1/4.3^99
⇒4A<3/4-1/4.3^99
⇒A<(3/4-1/4.3^99):4
⇒A<3/16-1/16.3^99<1/2 Vậy 1/3-2/3^2+3/3^3-4/3^4+...+99/3^99-100/3^100<3/16
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