Ta có \(3 S = 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}}\).

\(3 S - S = \left(\right. 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}} \left.\right) - \left(\right. \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \ldots . . + \frac{1}{3^{2 021}} + \frac{1}{3^{2 022}} \left.\right)\)

\(= 1 - \frac{1}{3^{2 022}}\)

\(3 S - S = 1 - \frac{1}{3^{2 022}}\)

\(S = \frac{1}{2} - \frac{1}{2. 3^{2 022}}\)

Vậy \(S < \frac{1}{2}\).

Ta có \(3 S = 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}}\).

\(3 S - S = \left(\right. 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}} \left.\right) - \left(\right. \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \ldots . . + \frac{1}{3^{2 021}} + \frac{1}{3^{2 022}} \left.\right)\)

\(= 1 - \frac{1}{3^{2 022}}\)

\(3 S - S = 1 - \frac{1}{3^{2 022}}\)

\(S = \frac{1}{2} - \frac{1}{2. 3^{2 022}}\)

Vậy \(S < \frac{1}{2}\).

Ta có \(3 S = 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}}\).

\(3 S - S = \left(\right. 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}} \left.\right) - \left(\right. \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \ldots . . + \frac{1}{3^{2 021}} + \frac{1}{3^{2 022}} \left.\right)\)

\(= 1 - \frac{1}{3^{2 022}}\)

\(3 S - S = 1 - \frac{1}{3^{2 022}}\)

\(S = \frac{1}{2} - \frac{1}{2. 3^{2 022}}\)

Vậy \(S < \frac{1}{2}\).

Ta có \(3 S = 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}}\).

\(3 S - S = \left(\right. 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}} \left.\right) - \left(\right. \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \ldots . . + \frac{1}{3^{2 021}} + \frac{1}{3^{2 022}} \left.\right)\)

\(= 1 - \frac{1}{3^{2 022}}\)

\(3 S - S = 1 - \frac{1}{3^{2 022}}\)

\(S = \frac{1}{2} - \frac{1}{2. 3^{2 022}}\)

Vậy \(S < \frac{1}{2}\).

Ta có \(3 S = 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}}\).

\(3 S - S = \left(\right. 1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . + + \frac{1}{3^{2 021}} \left.\right) - \left(\right. \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \ldots . . + \frac{1}{3^{2 021}} + \frac{1}{3^{2 022}} \left.\right)\)

\(= 1 - \frac{1}{3^{2 022}}\)

\(3 S - S = 1 - \frac{1}{3^{2 022}}\)

\(S = \frac{1}{2} - \frac{1}{2. 3^{2 022}}\)

Vậy \(S < \frac{1}{2}\).