Question

设函数f(x)=x的m次方+ax的导函数f’(x)=2x+1，则数列｛1/f(n)｝(n∈N*)的前项和是

Answer 1

f(x)=x²+x,1/f(n)=1/(n*(n+1))=1/n-1/(n+1)
1/f1+1/f2+.........1/f(n)=1-1/(n+1)=n/(n+1).

Answer 2

f'(x)=mx^(m-1)+a=2x+1;
m=2;
f(x)=x^2+x
1/f(n)=1/(n^2+n)=1/2n-1/2(n+2);
前n项和sn=1/2-1/6+1/4-1/8+1/6-1/10...+1/(2n-2)-1/2(n+1)+1/2n-1/2(n+2)=1/2*[1+1/2-1/(n+1)-1/(n+2)]=3/2-1/(n+1)-1/(n+2)