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Which is the integer part of the number N = 1/(1*2)+1/(2*3)+1/(3*4)+.....+1/(2007*2008)?

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cartoffried | Student | eNoter

Posted May 29, 2009 at 9:59 PM via web

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Which is the integer part of the number

N = 1/(1*2)+1/(2*3)+1/(3*4)+.....+1/(2007*2008)?

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neela | High School Teacher | Valedictorian

Posted May 30, 2009 at 2:36 AM (Answer #1)

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Given N= 1/(1*2)+1/(2*3)+(1/3*4)+.......+1/2007*2008.

Each term on the right is of the form 1/(r*(r+1))=1/r-1/(r+1).

Therefore,splitting each term in this form we get,

N=(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+...+(1/2006-1/2007)+(1/2007-1/2008).

Rearranging we get,

N=1/1+(-1/2+1/2)+(-1/3+1/3)+(-1/4+1/4)+...(-1/2006+1/2006)+(-1/2007+1/2007)+(-1/2008)

=1+0+0+0+......+0+(-1/2008)

=1-1/2008

= 2007/2008

=0.999501992..

Therefore,the integer part of the N is 0

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