1/1.2+1/2.3+.......+1/99.100.... Question from @gogo21

August 2018 2 55 Report

1/1.2+1/2.3+.......+1/99.100

acim Telescopic :
$\frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + .... + \frac{1}{98 . 99} + \frac{1}{99 . 100} \\ \\ = (\frac{1}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+ .... +(\frac{1}{98}-\frac{1}{99})+(\frac{1}{99}-\frac{1}{100}) \\ \\ = (\frac{1}{1}-\frac{1}{100}) \\ \\ = (\frac{100}{100}-\frac{1}{100}) \\ \\ = \frac{99}{100} \\ \\ = 0,99$

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Shadow $\frac{1}{1.2}$ + $\frac{1}{2.3}$ + ... + $\frac{1}{99.100}$
= $\frac{1}{1}$ - $\frac{1}{2}$ + $\frac{1}{2}$ - $\frac{1}{3}$ + ... + $\frac{1}{99}$ - $\frac{1}{100}$
= $\frac{1}{1}$ - $\frac{1}{100}$
= $\frac{99}{100}$
= 0.99

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