Calculate the expression

E=sqrt(1 + 1/2^2 + 1/3^2)+sqrt(1 + 1/3^2 + 1/4^2)+...+sqrt(1 + 1/n^2 + 1/(n+1)^2).

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For solving this exercise, we'll use the following way of approaching:

a^2+b^2+c^2=(a+b+c)^2-2(ab+ac+bc)

1+1/n^2+1/(n+1)^2=(1+1/n-1/(n+1))^2-2[1/n -1/(n+1)-

1/n*(n+1)]=(1+1/n-1/(n+1))^2-2(n+1-n-1)/n*(n+1)=

=(1+1/n-1/(n+1))^2 - 2*0=1+1/n-1/(n+1))^2

E=(1+1/2-1/3)+(1+1/3-1/4)+....+(1+1/(n-1)-1/n)+

(1+1/n-1/(n+1))=n-1+1/2-1/(n+1)=n-1/2-1/(n+1)=

=[2n(n+1)-(n+1)-2]/2(n+1)=(2n^2+2n-n-1-2)/2(n+1)=

=(2n^2+n-2)/2(n+1)

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