Question

若f(x)=a^x/(a^x+√a),求f(1/1001)+f(2/1001)+f(3/1001)+……+f(1000/

Answer 1

f(x)=a^x/(a^x+√a)

f(1-x)=a^(1-x)/[a^(1-x)+√a]

=a/(a+√a*a^x)

=√a/(a^x+√a)

f(x)+f(1-x)=a^x/(a^x+√a)+√a/(a^x+√a)=1

f(1/1001)+f(2/1001)+f(3/1001)+……+f(1000/1001)

=[f(1/1001)+f(1000/1001)]+[f(2/1001)+f(999/1001)]+……+[f(500/1001)+f(501/1001)]

=500