Răspuns:
[tex] \frac{1}{3 } + \frac{1}{303} + \frac{1}{3003} + \frac{1}{30003} = \frac{1}{3 \times 10 + 3} + \frac{1}{3 \times 100 + 3} + \frac{1}{3 \times 1000 + 3} + \frac{1}{3x10000 + 3} = [/tex]
[tex] \frac{1}{55} + \frac{1}{505} + \frac{1}{5005 } + \frac{1}{50005} = \frac{1}{5 \times 10 + 5} + \frac{1} {5 \times 100 + 5} \times \frac{1}{5 \times 1000 + 5} + \frac{5}{5 \times 10000 + 5} = [/tex]
[tex] = \frac{1}{3(10 + 1)} + \frac{1}{3(100 + 1)} + \frac{1}{3(1000 + 1)} + \frac{1}{3(10000 + 1)} = [/tex]
[tex] = \frac{1}{5(10 + 1)} + \frac{1}{5(100 + 1)} + \frac{1}{5(1000 + 1)} + \frac{1}{5(10000 + 1)} = [/tex]
[tex] = \frac{1}{3 \times 11} + \frac{1}{3 \times 101} + \frac{1}{3 \times 1001} + \frac{1}{3 \times 10001} = [/tex]
[tex] \frac{1}{5 \times 11} + \frac{1}{5 \times 101} + \frac{1}{5 \times 1001} + \frac{1}{5 \times 10001} = [/tex]
[tex] \frac{1}{3} \times ( \frac{1}{11} + \frac{1}{101} + \frac{1}{1001} + \frac{1}{1001} ) = [/tex]
[tex] \frac{1}{5} \times ( \frac{1}{11} + \frac{1}{101} + \frac{1}{1001} + \frac{1}{10001} ) = [/tex]
[tex] \frac{1}{3} \\ \frac{1}{5} = \frac{1}{3} \times \frac{1}{5} = \frac{5}{3} [/tex]
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