Tính E = 1/10 + 1/100 + 1/1000 + 1/10000 + 1/100000 + 1/1000000
Câu hỏi:
Tính \(E = \frac{1}{{10}} + \frac{1}{{100}} + \frac{1}{{1000}} + \frac{1}{{10000}} + \frac{1}{{100000}} + \frac{1}{{1000000}}\).
Trả lời:
\(E = \frac{1}{{10}} + \frac{1}{{100}} + \frac{1}{{1000}} + \frac{1}{{10000}} + \frac{1}{{100000}} + \frac{1}{{1000000}}\)
\(E = \frac{1}{{{{10}^1}}} + \frac{1}{{{{10}^2}}} + \frac{1}{{{{10}^3}}} + \frac{1}{{{{10}^4}}} + \frac{1}{{{{10}^5}}} + \frac{1}{{{{10}^6}}}\)
Suy ra: \(10E = 1 + \frac{1}{{{{10}^1}}} + \frac{1}{{{{10}^2}}} + \frac{1}{{{{10}^3}}} + \frac{1}{{{{10}^4}}} + \frac{1}{{{{10}^5}}}\)
10E – E = \(1 + \frac{1}{{{{10}^1}}} + \frac{1}{{{{10}^2}}} + \frac{1}{{{{10}^3}}} + \frac{1}{{{{10}^4}}} + \frac{1}{{{{10}^5}}} - \frac{1}{{{{10}^1}}} - \frac{1}{{{{10}^2}}} - \frac{1}{{{{10}^3}}} - \frac{1}{{{{10}^4}}} - \frac{1}{{{{10}^5}}} - \frac{1}{{{{10}^6}}}\)
9E = \(1 - \frac{1}{{{{10}^6}}} = \frac{{{{10}^6} - 1}}{{{{10}^6}}}\)
Suy ra: E = \(\frac{{{{10}^6} - 1}}{{{{9.10}^6}}}\).
