# Answer to Question #16744 in Real Analysis for Jay

Question #16744

n(-N: (1/1)*2+(1/2)*3+...+(1/n)(n+1)=n/(n+1)

Expert's answer

1/(k*(k+1))=1/k-1/(k+1)

so

1/(1*2)+1/(2*3)+....+1/(n*(n+1))=1-1/2+1/2-1/3+...+1/(n-1)-1/n+1/n-1/(n+1)=1-1/(n+1)=n/(n+1)

so

1/(1*2)+1/(2*3)+....+1/(n*(n+1))=1-1/2+1/2-1/3+...+1/(n-1)-1/n+1/n-1/(n+1)=1-1/(n+1)=n/(n+1)

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