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The sum of the series (1)/(1.23)+(1)/(...

The sum of the series
`(1)/(1.23)+(1)/(3.45)+(1)/(5.67)+…infty` is

A

`log_(e )2-1/2`

B

`log_(e)2`

C

`log_(e )2+1/2`

D

`log_(e )2+1`

Text Solution

Verified by Experts

We have
`(1)/(1.2.3)+(1)/(3.4.5)+(1)/(5.6.7)+…infty`
`=underset(n=1)overset(infty)Sigma(1)/((2n-1)2n(2n+1))`
`underset(n=1)overset(infty)Sigma{(1)/(2(2n-1))-(1)/(2n) (1)/(2(2n+1))}`
`=1/2underset(n=1)overset(infty)Sigma{(1)/(2n-1)-(2)/(2n)+(1)/(2n+1)}`
`=1/2 underset(n=1)overset(infty)Sigma{((1)/(2n-1)-(1)/(2n))-((1)/(2n)-(1)/(2n+1))}`
`=1/2log_(e)2+1/2log_(e)2-1/2=log_(e)2-1/2`
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OBJECTIVE RD SHARMA-EXPONENTIAL AND LOGARITHMIC SERIES-Chapter Test
  1. The sum of the series (1)/(1.23)+(1)/(3.45)+(1)/(5.67)+…infty is

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  2. The series expansion of log{(1+x)^(1+x)(1-x)^(1-x)} is

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  3. 2log x-log(x+1)-log(x-1) is equals to

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  4. The coefficient of x^(n) in the expansion of log(e)(1)/(1+x+x^(2)+x^...

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  5. If x ne 0 then the sum of the series 1+(x)/(2!)+(2x^(2))/(3!)+(3x^(3...

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  6. If log(1-x+x^(2))=a(1)x+a(2)x^()2)+a(3)x^(2)+a(3)x^(3)+…and n is not a...

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  7. If log(1-x+x^(2))=a(1)x+a(2)x^(2)+a(3)x^(3)+… then a(3)+a(6)+a(9)+.....

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  8. The coefficient of x^(n) in the expansion of log(a)(1+x) is

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  9. The coeffiecent of n^(-r) in the expansion of log(10)((n)/(n-1)) is

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  10. The sum of the series (x-1)/(x+1)+1/2(x^(2)-1)/((x+1)^(2)+1/3(x^(3)-...

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  11. The sum of series 2[ 7^(-1)+3^(-1).7^(-3)+5^(-1).7^(-5)+...] is

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  12. The coefficient of x^(6) in the expansion of log{(1+x)^(1+x)(1-x)^(...

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  13. The sum of the series 1/2x^2+2/3x^3+3/4x^4+4/5x^5+... is :

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  14. If x,y,z are three consecutive positive integers and X-Z + 2 = 0, then...

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  15. The sum of the series ((1)^(2).2)/(1!)+(2^(2).3)/(2!)+(3^(2).4)/(3!)+(...

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  16. The value of 1-log(e)2+(log(e)2)^(2)/(2!)-(log(e)2)^(3)/(3!)+.. is

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  17. 1+(loge n)^2 /(2!) + (loge n )^4 / (4!)+...=

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  18. (2)/(3!)+(4)/(5!)+(6)/(7!)+..is equal to

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  19. Sum of n terms of the series 1/(1.2.3.4.)+1/(2.3.4.5) +1/(3.4.5.6)+.....

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  20. The value of 1+(log(e)x)+(log(e)x)^(2)/(2!)+(log(e)x)^(3)/(3!)+…inft...

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  21. If |x|lt1 then the coefficient of x^(3) in the expansion of log(1+x+x^...

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