(1)/(5)+(1)/(2.5^(2))+(1)/(3.5^(3))+(1)/...

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

`log_(e)((8)/(e))`

`log_(e)((2)/(e))`

`log_(e)((5)/(4))`

`2-log_(e)2`

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(1)/(1.3).(1)/(2)+(1)/(2.4).(1)/(2^(2))+(1)/(3.5).(1)/(2^(3))+....=

(1)/(1.2)-(1)/(2.3)+(1)/(3.4)-(1)/(4.5)+....=

If (3)/(4)+(1)/(3)((3)/(4))^(3)+(1)/(5)((3)/(4))^(5)+....=log_(e)a,(1)/(3)+(1)/(3.3^(3))+(1)/(5.3^(5))+(1)/(7.3^(7))+....=log_(e)b, 1+(1)/(3.2^(2))+(1)/(5.2^(4))+(1)/(7.2^(6))+....=log_(e)c then the ascending order of a, b, c is

Assertion (A) : (1)/(5)+(1)/(3.5^(3))+(1)/(5.5^(5))+(1)/(7.5^(7))+…(1)/(2)log((3)/(2)) Reason (R ) : If |x| lt 1 then log_(e )((1+x)/(1-x))=2(x+(x^(3))/(3)+(x^(5))/(5)+…)

x=(1)/(3)+(1)/(3.3^(3))+(1)/(5.3^(5))+….. y=(1)/(5)+(1)/(3.5^(3))+(1)/(5.5^(5))+…. z=1+(1)/(3.2^(2))+(1)/(5.2^(4))+(1)/(7.2^(6))+….. Then descending order of x, y, z

log2+2[(1)/(5)+(1)/(3.5^(3))+(1)/(5.5^(5))+….oo]=

The sum of the series (1)/(2)((1)/(5))^(2)+(2)/(3)((1)/(5))^(3)+(3)/(4)((1)/(5))^(4)+... 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)/(2.3)+(1)/(4.5)+(1)/(6.7)+……oo=

(1)/(2)-(1)/(3(1!))+(1)/(4(2!))-(1)/(5(3!))+....=

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

(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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e^(x-1-(1)/(2)(x-1)^(2)+(1)/(3)(x-1)^(3)-(1)/(4)(x-1)^(4))+...=
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If S=((b-1)-(1)/(2)(b-1)^(2)+(1)/(3)(b-1)^(3)-....)/((a-1)-(1)/(2)(a-1...
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(1)/(x+1)+(1)/(2(x+1)^(2))+(1)/(3(x+1)^(3))+....=
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(x)/(x+1)+(1)/(2)((x)/(x+1))^(2)+(1)/(3)((x)/(x+1))^(3)+....=
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(2ax)/(a^(2)+x^(2))+(1)/(3)((2ax)/(a^(2)+x^(2)))^(3)+(1)/(5)((2ax)/(a^...
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((2n)/(n^(2)+1))+(1)/(3)((2n)/(n^(2)+1))^(3)+(1)/(5)((2n)/(n^(2)+1))^(...
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(1)/(2x-1)+(1)/(3).(1)/((2x-1)^(3))+(1)/(5)(1)/((2x-1)^(5))+....=
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(1)/(2x+1)+(1)/(3)(1)/((2x+1)^(3))+(1)/(5)(1)/((2x+1)^(5))+....=
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(1)/(2n^(2)-1)+(1)/(3(2n^(2)-1)^(3))+(1)/(5(2n^(2)-1)^(5))+....=
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If xgt0 then (x-1)/(x+1)+(1)/(2)(x^(2)-1)/((x+1)^(2))+(1)/(3)(x^(3)-1)...
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log(1+x+x^(2)+...oo)=
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