Question-126097

Question Number 126097 by amns last updated on 17/Dec/20

Commented by amns last updated on 17/Dec/20

I need {someone}^{'} s help to solve it!!!

Answered by Dwaipayan Shikari last updated on 17/Dec/20

\frac{1}{2} - \frac{1}{12} - \frac{1}{20} - \frac{1}{30} - \frac{1}{42} - . . \frac{1}{210} = 1 - (\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \dots) + \frac{1}{6}

= 1 - \underset{n = 1}{\sum^{14}} \frac{1}{n} - \frac{1}{n + 1} + \frac{1}{6} = 1 - (1 - \frac{1}{15}) + \frac{1}{6} = \frac{7}{30}

N o t e : \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \dots + \frac{1}{210}

= 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + \dots + \frac{1}{14} - \frac{1}{15}

= 1 - \frac{1}{15} = \frac{14}{15}