log(e)2=...

`log_(e)2=`

`1+(1)/(2)+(1)/(3)+(1)/(4)+....`

`1-(1)/(2)-(1)/(3)-(1)/(4)-....`

`(1)/(1.2)+(1)/(3.4)+(1)/(5.6)+....`

`(1)/(1.2)+(1)/(2.3)+(1)/(3.4)+....`

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If log_(e )k=2[(3)/(5)+(1)/(3)((3)/(5))^(3)+(1)/(5)((3)/(5))^(5)+….oo] then k =

Observe the following lists. {:("LIST - I","LIST - II"),((A) 8^(th)" term of log"_(e)^(2) " is",(1)log((n+1)/(n-1))),((B)7^(th)" term of log"_(e )(5//4)" is",(2)(-1)/(8)),((C )n^(th)" term of log"_(e )(3//2)" is",(3)(1)/(7.4^(7))),((D)(2)/(n^(2)+1)+(1)/(3)((2n)/(n^(2)+1))^(3)+.....,(4)( (-1)^(n-1))/(n.2^(n))):} The correct match for List-I from List-II is

Observe the following lists {:("List-I","List-II"),((A) 1-(1)/(2)+(1)/(3)-(1)/(4)+(1)/(5)+...oo,(1)(2)/(3)+ (1)/(2)log2),((B)(1)/(5)+(1)/(2.5^(2))+(1)/(3.5^(3))+...oo,(2)log_(e )2),((C )(1)/(n+1)+(1)/(2(n+1)^(2))+(1)/(3(n+1)^(3))...oo,(3)-log_(e )((4)/(5))),((D)1+(1)/(3.3^(3))+(1)/(5.3^(5))+...oo,(4)log_(e )((5)/(4))),(,(5)-log_(e )(1-(1)/(n+1))):} The correct match for List - I from List -II is

1+2(log_(e)a)+(2^(2))/(2!)(log_(e)a)^(2)+(2^(3))/(3!)(log_(e)a)^(3)+....=

2[1+((log_(e)a)^(2))/(2!)+((log_(e)a)^(4))/(4!)+....] =

A : log_(e)3+((log_(e)3)^(2))/(2!)+((log_(e)3)^(3))/(3!)+....=2 R : If x,ainRandagt0" then "a^(x)=1+xlog_(e)a+(x^(2))/(2!)(log_(e)a)^(2)+(x^(3))/(3!)(log_(e)a)^(3)+....

log_(e)3+((log_(e)3)^(2))/(2!)+((log_(e)3)^(3))/(3!)+....=

The 4^(th) term of "log"_(e )(3)/(2) is

DIPTI PUBLICATION ( AP EAMET)-EXPONENTIAL SERIES & LOGARITHMIC SERIES (APPENDIX-1)-EXERCISE 1B

1+(1)/(3.2^(2))+(1)/(5.2^(4))+(1)/(7.2^(6))+....=
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log(4)2-log(8)2+log(16)2-....=
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log(e)2=
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(1)/(1.2)+(1)/(3.4)+(1)/(5.6)+....
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(1)/(2.3)+(1)/(4.5)+(1)/(6.7)+(1)/(8.9)+...
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(1)/(1.2)-(1)/(2.3)+(1)/(3.4)-(1)/(4.5)+....=
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(1)/(1.3)+(1)/(2.5)+(1)/(3.7)+(1)/(4.9)+...=
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(1)/(1.2.3)+(1)/(3.4.5)+(1)/(5.6.7)+....
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(5)/(1.2.3)+(7)/(3.4.5)+(9)/(5.6.7)+....=
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The sum of the series (1)/(1.3.5)+(1)/(3.5.7)+(1)/(5.7.9)+... is
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Sum of the series (1)/(4.5.6)+(1)/(5.6.7)+(1)/(6.7.8)+... upto oo is
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(1)/(1.3)+(1)/(2)((1)/(3.5))+(1)/(3)((1)/(5.7))+....=
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(2)/(3)+(1)/(2)((2)/(3))^(2)+(1)/(3)((2)/(3))^(3)+....=
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(1)/(5)+(1)/(2.5^(2))+(1)/(3.5^(3))+(1)/(4.5^(4))+.......=
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The sum of the series (1)/(2)((1)/(5))^(2)+(2)/(3)((1)/(5))^(3)+(3)/(4...
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(1)/(2)-(1)/(2).(1)/(2^(2))+(1)/(3).(1)/(2^(3))-(1)/(4).(1)/(2^(4))+.....
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(1)/(2)+(3)/(2).(1)/(4)+(5)/(3).(1)/(8)+(7)/(4).(1)/(16)+....=
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Sum of the series (1)/(2)((1)/(2)+(1)/(3))-(1)/(4)((1)/(2^(2))+(1)/(3^...
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(a-b)/(a)+(1)/(2)((a-b)/(a))^(2)+(1)/(3)((a-b)/a)^(3)+....=
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(a-1)-(1)/(2)(a-1)^(2)+(1)/(3)(a-1)^(3)-(1)/(4)(a-1)^(4)+...=
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