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log en -log e (n-1) =(1)/(n) +(1)/( 2n ^...

`log _en -log _e (n-1) =(1)/(n) +(1)/( 2n ^(2) ) +(1)/( 3n ^(3))+(1)/( 4n ^(4)).........infty `

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SHIVLAAL PUBLICATION-परिशिष्ट :अनंत श्रेणियाँ-प्रश्नावली A.5
  1. log e ""(3)/(2) =2( (1)/(5) +( 1)/( 3.5^(3))+(1)/(5.5^(5))+......infty...

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  2. log e""3=1 +(1)/( 3.2^(2) )+( 1)/(5.2^(4))+(1)/( 7.2^(6))+......... अन...

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  3. loge ""( 4)/(3) =2 {(1)/(7)+ (1)/(3) ( 1)/( 7^(3))+(1)/(5) (1)/( 7^(5...

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  4. loge ""(5)/(4) =(1)/(5) +(1)/( 2.5^(2)) +(1)/( 3.5^(3)) +(1)/( 4.5^(4)...

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  5. log e ""2 =(4)/( 1.3) -( 6)/( 2.4) +(12)/( 5.7) -(14)/( 6.8) + ..........

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  6. log e 4=1 +(2)/( 1.2.3) +(2)/( 3.4.5) +( 2)/( 5.6.7) +.......infty

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  7. 2-log e 2""=(1)/(2) +(3)/( 2).(1)/(4) +(5)/(3) .(1)/(8) +( 7)/(4) .(1)...

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  8. loge (n+1) -log e n=(1)/(n) -(1)/( 2n ^(2) )+ ( 1)/(3n ^(3))-(1)/( 4n ...

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  9. log en -log e (n-1) =(1)/(n) +(1)/( 2n ^(2) ) +(1)/( 3n ^(3))+(1)/( 4n...

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  10. log e ""(n+1)/(n-1) =2 ((1)/(n)+ ( 1)/(3n ^(3))+(1)/( 5n ^(5))+(1)/( 7...

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  11. (1)/(3) + ( 1)/( 3.3^(3)) +(1)/( 5.3^(5))+(1)/( 7.3^(7)).......infty =...

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  12. log e (n+1) -log e n =2 ""{(1)/( 2n +1) + ( 1)/( 3(2n + 1)^(3))+(1)/( ...

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  13. log e ""(a+ x)/(a-x) =( 2ax) /(a^(2) +x^(2))+(1)/(3) ((2ax)/(a^(2) +x^...

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  14. log e (1+(1)/( n ))^(n) =1- (1)/(2( n +1) )- ( 1)/( 2.3(n+1) ^(2))-(1)...

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  15. log e x =(x-1)/(x+1) +(1)/(2) (x^(2)-1)/( (x+1) ^(2))+(1)/(3) (x^(3) -...

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  16. (x^(2))/(2) +(2x^(3))/( 3) +( 3x^(4))/( 4) +( 4x^(5))/(5)+..........=(...

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  17. (1)/( log e2)=1+(log e 2 )/(|2) +((log e2)^(2))/( |3) +( (log e 2)^(3)...

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  18. log e (1+2x +3x^(2) +4x^(3)+............) =2 (x+(x^(2))/( 2) +(x^(3))/...

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  19. यदि y=x+ (x^(2))/(2) +(x^(3))/( 3) +( x^(4))/(4) +.........infty तो ...

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  20. यदि |x| lt 1 तो सिद्ध कीजिए की : log [(1+x) ^(1+ x) (1-x) ^(1-x) ...

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