log2=(1)/(1.2)+(1)/(3.4)+(1)/(5.6)+...

If x = (1)/(1.2) + (1)/(3.4) + (1)/(5.6) …….oo, and y = 1 - (1)/(2.3) - (1)/(4.5) - (1)/(6.7)…….oo , then

If x = (1)/(1.2) + (1)/(3.4) + (1)/(5.6) …….oo, and y = 1 - (1)/(2.3) - (1)/(4.5) - (1)/(6.7)…….oo , then

Show that, log_e2=1/(1.2)+1/(3.4)+1/(5.6)+.....infty

I:(1)/(1.2)+(1)/(3.4)+(1)/(5.6)+....log_(e)2 II:(1)/(1.2)-(1)/(2.3)+(1)/(3.4)-(1)/(4.5)+....=2log_(e)2-1

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

The series expansion of log[(1 + x)^((1 + x))(1-x)^(1-x)] is (1) 2[(x^(2))/(1.2) + (x^(4))/(3.4)+(x^(6))/(5.6)+...] (2) [(x^(2))/(1.2) + (x^(4))/(3.4)+(x^(6))/(5.6)+...] (3) 2[(x^(2))/(1.2) + (x^(4))/(2.3)+(x^(6))/(3.4)+...] (4) 2[(x^(2))/(1.2) -(x^(4))/(2.3)+(x^(6))/(3.4)-...]

The series expansion of log_(e) [(1 + x^((1 + x))(1-x)^(1-x)] is (1) 2[(x^(2))/(1.2) + (x^(4))/(3.4)+(x^(6))/(5.6)+...] (2) [(x^(2))/(1.2) + (x^(4))/(3.4)+(x^(6))/(5.6)+...] (3) 2[(x^(2))/(1.2) + (x^(4))/(2.3)+(x^(6))/(3.4)+...] (4) 2[(x^(2))/(1.2) -(x^(4))/(2.3)+(x^(6))/(3.4)-...]

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