Ta có:
` D = 1/3 + \frac{1}{3^{2}} + \frac{1}{3^{3}} + ... + \frac{1}{3^{100}} `
` <=> 3D = 1 + 1/3 + \frac{1}{3^{2}} + ... + \frac{3^{99}} `
` <=> 3D - D = ( 1 + 1/3 + \frac{1}{3^{2}} + ... + \frac{1}{3^{99}} ) - ( 1/3 + \frac{1}{3^{2}} + \frac{1}{3^{3}} + ... + \frac{1}{3^{100}} ) `
` <=> 2D = 1 + 1/3 + \frac{1}{3^{2}} + ... + \frac{3^{99}} - 1/3 - \frac{1}{3^{2}} - \frac{1}{3^{3}} - ... - \frac{1}{3^{100}} `
` <=> 2D = 1 + (1/3 - 1/3) + ( \frac{1}{3^{2}} - \frac{1}{3^{2}} ) + ... + (\frac{1}{3^{99}} - \frac{1}{3^{99}}) - \frac{1}{3^{100}} `
` <=> 2D = 1 - \frac{1}{3^{100}} `
` <=> D = \frac{1 - \frac{1}{3^{100}}}{2} `
` <=> D = 1/2 - \frac{1}{2.3^{100}} `
Do: ` 1/2 - \frac{1}{2.3^{100}} < 1/2 `
` => D < 1/2 ` ` (đpcm) `
