а)[tex] \frac{1}{1 \times 2}+\frac{1}{2 \times 3} + ... + \frac{1}{19 \times 20} = \sum{n=1}^{19} \frac{1}{n(n+1)} = \sum{n=1}^{19} \left(\frac{1}{n} - \frac{1}{n+1}\right) = 1 - \frac{1}{20} = \frac{19}{20} [/tex]
б)[tex] \frac{1}{10 \times 11} + \frac{1}{11 \times 12} + ... + \frac{1}{19 + 20} = \sum{n=10}^{19} \frac{1}{n(n+1)} = \sum{n=10}^{19} \left(\frac{1}{n} - \frac{1}{n+1}\right) = \frac{1}{10} - \frac{1}{20} = \frac{1}{10} [/tex]
в)[tex] \frac{1}{1 \times 4} + \frac{1}{4 \times 7} + ... + \frac{1}{13 \times 16} = \sum{n=1}^{13} \frac{1}{3n-2(3n-1)} = \sum{n=1}^{13} \left(\frac{1}{3n-2} - \frac{1}{3n-1}\right) = \frac{1}{1} - \frac{1}{39} = \frac{38}{39} [/tex]
г)[tex] \frac{1}{3 \times 7} + \frac{1}{7 \times 11} + ... + \frac{1}{19 \times 23} = \sum{n=1}^{17} \frac{1}{4n-1(4n+3)} = \sum{n=1}^{17} \left(\frac{1}{4n-1} - \frac{1}{4n+3}\right) = \frac{1}{3} - \frac{1}{71} = \frac{68}{71} [/tex]
а)
[tex] \frac{1}{1 \times 2}+\frac{1}{2 \times 3} + ... + \frac{1}{19 \times 20} = \sum{n=1}^{19} \frac{1}{n(n+1)} = \sum{n=1}^{19} \left(\frac{1}{n} - \frac{1}{n+1}\right) = 1 - \frac{1}{20} = \frac{19}{20} [/tex]
б)
[tex] \frac{1}{10 \times 11} + \frac{1}{11 \times 12} + ... + \frac{1}{19 + 20} = \sum{n=10}^{19} \frac{1}{n(n+1)} = \sum{n=10}^{19} \left(\frac{1}{n} - \frac{1}{n+1}\right) = \frac{1}{10} - \frac{1}{20} = \frac{1}{10} [/tex]
в)
[tex] \frac{1}{1 \times 4} + \frac{1}{4 \times 7} + ... + \frac{1}{13 \times 16} = \sum{n=1}^{13} \frac{1}{3n-2(3n-1)} = \sum{n=1}^{13} \left(\frac{1}{3n-2} - \frac{1}{3n-1}\right) = \frac{1}{1} - \frac{1}{39} = \frac{38}{39} [/tex]
г)
[tex] \frac{1}{3 \times 7} + \frac{1}{7 \times 11} + ... + \frac{1}{19 \times 23} = \sum{n=1}^{17} \frac{1}{4n-1(4n+3)} = \sum{n=1}^{17} \left(\frac{1}{4n-1} - \frac{1}{4n+3}\right) = \frac{1}{3} - \frac{1}{71} = \frac{68}{71} [/tex]