Ta có:
\(M=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{99}}\)
\(\Rightarrow3M=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{98}}\)
\(\Rightarrow3M-M=\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{98}}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{99}}\right)\)
\(\Rightarrow2M=1-\frac{1}{3^{98}}\)
\(\Rightarrow M=\left(1-\frac{1}{3^{98}}\right):2\)
\(\Rightarrow M=\frac{1}{2}-\frac{1}{3^{98}.2}< \frac{1}{2}\)
\(\Rightarrow M< \frac{1}{2}\left(đpcm\right)\)
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