Evaluate the series

Question:
If !!f(x)!! !!=!! !![9^x/(9^x + 3)]!!
Then !!f(1/1996) + f(2/1996) + .......+f(1995/1996)!! !!=!! !!?!!
!!a)" "996.5!!

!!b)" "997.5!!

!!c)" "998.5!!

!!d)" "999.5!!

Solution:
We have

!!f(x)!! !!=!! !![9^x/(9^x + 3)]!!

and !!f(1-x)!! !!=!! !![3/(3 + 9^x)]!!

so that

!!f(x)!! !!+!! !!f(1-x)!! !!=!! !!1!!

Therefore
!![f(1/1996) + f(1995/1996)]!! !!=!! !!1!!

!![f(2/1996) + f(1994/1996)]!! !!=!! !!1!!

.... .... .... ....

!!f(m-1/1996) + f(m+1/1996)!! !!=!! !!1!!

Here m is the middle term of this sequence

!!=!! !![(1 + 1995)/2.1996]!! !!=!! !!1/2!!

and !!f(1/2)!! !!=!! !!1/2!!

Therefore
!![f(1/1996) + f(2/1996) + ...... +f(1/2) + ... + f(1995/1996)]!! !!=!! !!(1 + 1 + .........upto 997 + 0.5 )!! !!=!! !!997.5!!

Get it on Google Play