b: \(M=\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\cdots+\frac{1}{1+2+\cdots+59}\)
\(=\frac{1}{3\cdot\frac42}+\frac{1}{4\cdot\frac52}+\cdots+\frac{1}{59\cdot\frac{60}{2}}=\frac{2}{3\cdot4}+\frac{2}{4\cdot5}+\cdots+\frac{2}{59\cdot60}\)
\(=2\left(\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\cdots+\frac{1}{59\cdot60}\right)\)
\(=2\left(\frac13-\frac14+\frac14-\frac15+\cdots+\frac{1}{59}-\frac{1}{60}\right)=2\left(\frac13-\frac{1}{60}\right)<2\cdot\frac13=\frac23\) (ĐPCM)
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