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The sumof the series1/(1*2*3)+1/(3*4*5)+1/(5*6*7)+... is |
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Answer» ` log_(e)2-1/2` ` :. T_(n) = 1/((2n-1)(2n)(2n+1))` ` = 1/(2n-1)=1/(2n) +1/(2(2n+1))` ` = 1/2 [ 1/(2n-1)-1/(2n)] -1/2 [ 1/(2n)- 1/(2n+1)]` On PUTTING n =1,2,3,..., ` T_(1)= 1/2 [ 1/1 -1/2 ] -1/2 [ 1/2 -1/3]` ` T_(2) = 1/2 [ 1/3 - 1/4 ] - 1/2 [ 1/4 -1/5] ` `{:(".................."),(".................."):}` ` :. S = T_(1) +T_(2) +T_(3) +...+ T_(n) +...` ` =1/2 [ 1- 1/2 + 1/3 +1/4 +1/5 -1/6 +1/7-...]` ` -1/2 [ 1/2 - 1/3- 1/4 -1/5 +1/6 -1/7 +...]` ` -1/2 [1/2 -1/3 +1/4 -1/5 +1/6 -1/7 +...]` `= 1/2 log_(e)(1+1)+1/2 [ -1+{1-1/2 +1/3 -1/4 +...}]` ` = 1/2 log_(e) 2 - 1/2 +1/2 log_(e) (1+1) = log_(e) 2 - 1/2 ` |
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