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Thesum oftheseries(3)/(4.8)- (3.5)/(4.8.12) +(3.5.7)/(4.8.12.16) - … isequalto |
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Answer» `sqrt((3)/(2)) - (3)/(4) ` ` (3)/(4.8)- (3.5)/(4.8.12) +(3.5.7)/(4.8.12.16)- …. ` `=(3)/(2!4^2) - (3.5)/(4^3. 3!) + (3.5.7)/(4^4 4 ! )- … ` `( 1 +x ) ^n = 1+ (nx ) /(1!)+ ( n (n - 1 ))/(2!)x ^2 +(n(n - 1 ) (n - 2))/(3!)x ^ 3+ (n(n -1) (n - 2 ) (n - 3 ))/(4!) x ^ 4 +... ` On comparingtheaboveexpansion, weget `thereforen(n -1 ) x^ 2= (3)/(4^ 2 ) ""...(2)` `n ( n - 1 ) (n - 2 ) x ^3 = ( -3 (5))/(4^3)"" ...(3) ` ` n(n - 1)(n - 2 ) (n - 3 ) x ^ 4= ((3) (5)(7))/(4 ^ 4 ) "" `...(4) `(3)div(2) , (4) div (3) ` `rArr(n - 2 ) x =(-5 ) /(4) "" ...(5)` ` rArr(n - 3 ) x = ( -7) /(4 ) "" `...(6) `(5)- (6) ` ` x = (-5 )/(4 )+(7)/(4) ` `x =(1 )/(2) ` ` (n-2 ) x= ( -5 )/(4 ) ` ` (n -2 )(1)/(2)= (-5)/(4) ` ` (n - 2 )= ( -5)/(2) ` `rArrn =(- 1 ) /(2) ` `THEREFORE(1 + x ) ^n= 1+(nx ) /(1! )+ (n (n - 1 ))/(2! ) x ^ 2+(n(n - 1 ) (n - 2 ))/( 3 ! )x ^ 3 +... ` `(1 +(1)/(2)) ^(- (1)/(2)) = 1+ ((-(1)/(2) )(1/2))/(1!) ` ` + ((-(1)/(2))(-1/2 - 1))/(2!) ((1)/(2)) ^ 2 + ... ` ` sqrt(2/3)= 1 -(1 )/(4)+ (3 )/(4.8)- (3.5 )/(4.8.12)+ (3.5.7)/(4.8.12.16) - ... ` `therefore(3)/(4.8)- (3.5)/(4.8.12) + (3.5.7)/(4.8.12.16) - ... = sqrt(2/3)- (3)/(4) ` |
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