Let `g(x)=e^(f(x))a n df(x+1)=x+f(x)AAx in Rdot` If `n in I^+,t h e n(g^(prime)(n+1/2))/(g(n+1/2))-(g^(prime)(1/2))/(g(1/2))=` `2(1+1/2+1/3++1/n)` `2(1+1/3+1/5+1/(2n-1))` `n` 1

Options

A. `2(1+(1)/(2)+(1)/(3)+...+(1)/(n))`
B. `2(1+(1)/(3)+(1)/(5)+...+(1)/(2n-n))`
C. n
D. 1

Answer

C

Explanation

`f(x+1)=x+f(x)`
`rArr" "e^(f(x+1))=e^(x)+f(x)=e^(x)xxe^(f(x))`
`rArr" "g(x+1)=e^(x)xxg(x)`
`rArr" "lng(x+1))=x+ln(g(x))`
`rArr" "(g'(x+1))/(g(x+1))-(g'(x))/(g(x))=1`
`rArr" "(g'((1)/(2)+1))/(g((1)/(2)+1))-(g'((1)/(2)))/(g((1)/(2)))=1`
`{:(" "(g'((1)/(2)+2))/(g((1)/(2)+2))-(g'(1+(1)/(2)))/(g(1+(1)/(2)))=1),(" "overset(.)overset(.)." "overset(.)overset(.).),(" "(g'(n+(1)/(2)))/(g(n+(1)/(2)))-(g'(n-(1)/(2)))/(g(n-(1)/(2)))=1):}`
Adding `(g'(n+(1)/(2)))/(g(n+(1)/(2)))-(g'((1)/(2)))/(g((1)/(2)))=n`