Cho biểu thức $C$ = $\frac{1}{3}$ – $\frac{2}{3^{2}}$ + $\frac{3}{3^{3}}$ – $\frac{4}{3^{4}}$ + …….. + $\frac{99}{3^{99}}$ – $\frac{100}{3^{100}}$
Chứng minh rằng $C$ < $\frac{3}{16}$
Cho biểu thức $C$ = $\frac{1}{3}$ – $\frac{2}{3^{2}}$ + $\frac{3}{3^{3}}$ – $\frac{4}{3^{4}}$ + …….. + $\frac{99}{3^{99}}$ – $\frac{100}{3^{100}}$
Chứng minh rằng $C$ < $\frac{3}{16}$
`C= 1/3 – 2/3^2 + 3/3^3 – 4/3^4+…+99/3^99 – 100/3^100`
`3C= 3(1/3 – 2/3^2 + 3/3^3 – 4/3^4+…+99/3^99 – 100/3^100)`
`3C= 1 – 2/3 + 3/3^2 – 4/3^3 +…+ 99/3^98 – 100/3^99`
`3C+C = 1-2/3 + 3/3^2 – 4/3^3 +…+99/3^98 – 100/3^99 +1/3 – 2/3^2 + 3/3^3 – 5/3^4 +…+99/3^99 – 100/3^100`
`4C= 1- 1/3 + 1/3^2 – 1/3^3+….+1/3^98 – 1/3^99`
Đặt `A= 1 – 1/3 + 1/3^2 -1/3^3+…-1/3^99`
`1/3 A = 1/3( 1 – 1/3 + 1/3^2 -1/3^3+…-1/3^99)`
`1/3A = 1/3 – 1/3^2 + 1/3^3-….-1/3^100`
`A+ 1/3 A = 1 – 1/3 + 1/3^2 -1/3^3+…+1/3^98 – 1/3^99 + 1/3 -1/3^2 + 1/3^3 -….-1/3^100`
`4/3 A= 1- 1/3^100`
`A = (1-1/3^100) :4/3`
`A= 3/4 – 1/(3^99. 4)`
`=> 4C = (3/4 – 1/(3^99. 4))`
`=> C = (3/4 – 1/(3^99.4)) : 4`
`C= (3/4 – 1/(3^99. 4)) . 1/4`
`C= 3/16 – 1/(3^99. 16) < 3/16`
Vậy `C < 3/16`