\(\left(1+\frac{1}{2.3}\right)\left(1+\frac{1}{3.4}\right)\left(1+\frac{1}{4.5}\right)...\left(1+\frac{1}{99.100}\right)\)
\(=\left(1+\frac{1}{2}-\frac{1}{3}\right)\left(1+\frac{1}{3}-\frac{1}{4}\right)\left(1+\frac{1}{4}-\frac{1}{5}\right)...\left(1+\frac{1}{99}-\frac{1}{100}\right)\)
\(=1+\frac{1}{2}-\frac{1}{3}.1+\frac{1}{3}-\frac{1}{4}.1+\frac{1}{4}-\frac{1}{5}...1+\frac{1}{99}-\frac{1}{100}\)
\(=1+\frac{1}{2}-1.\left(\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-...-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\right)\)
\(=1+\frac{1}{2}-1.\left(2\frac{1}{3}-2\frac{1}{4}-...-2\frac{1}{99}-\frac{1}{100}\right)\)
\(=1+\frac{1}{2}-1\left[2.\left(\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-...-\frac{1}{99}\right)\right]-\frac{1}{100}\)
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