Question

How to calculate this sum? #S=(x+1/x)^2+(x^2+1/(x^2))^2+...+(x^n+1/(x^n))^2#.

Answer 1

#S=(x+1/x)^2+(x^2+1/(x^2))^2+...+(x^n+1/(x^n))^2#

#S=(x^2+x^4+ x^6+...+x^(2n))+(1/x^2+1/x^4+ 1/x^6+...+1/x^(2n))+2n#

#S=(x^2(x^(2n)-1))/(x^2-1) +((1/x^2(1-1/x^(2n)))/(1-1/x^2))+2n#

Taking #x>1#