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

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To calculate the sum ( S = (x + \frac{1}{x})^2 + (x^2 + \frac{1}{x^2})^2 + \ldots + (x^n + \frac{1}{x^n})^2 ), we can use the formula for the sum of squares of binomials:

[ (a + b)^2 = a^2 + 2ab + b^2 ]

For each term in the sum, let's denote ( a = x^k ) and ( b = \frac{1}{x^k} ), where ( k = 1, 2, \ldots, n ). Then, applying the formula, we have:

[ (x^k + \frac{1}{x^k})^2 = (x^k)^2 + 2(x^k)\left(\frac{1}{x^k}\right) + \left(\frac{1}{x^k}\right)^2 ] [ = x^{2k} + 2 + \frac{1}{x^{2k}} ]

So, each term in the sum simplifies to ( x^{2k} + 2 + \frac{1}{x^{2k}} ). Therefore, the sum ( S ) becomes:

[ S = (x^{2} + 2 + \frac{1}{x^{2}}) + (x^{4} + 2 + \frac{1}{x^{4}}) + \ldots + (x^{2n} + 2 + \frac{1}{x^{2n}}) ]

[ = x^{2} + x^{4} + \ldots + x^{2n} + 2n + \frac{1}{x^{2}} + \frac{1}{x^{4}} + \ldots + \frac{1}{x^{2n}} ]

[ = x^{2}(1 + x^{2} + \ldots + x^{2(n-1)}) + 2n + \frac{1}{x^{2}}(1 + x^{2} + \ldots + x^{2(n-1)}) ]

[ = x^{2}\left(\frac{x^{2n} - 1}{x^2 - 1}\right) + 2n + \frac{1}{x^{2}}\left(\frac{x^{2n} - 1}{x^2 - 1}\right) ]

[ = \frac{x^{2n+2} - x^2 + 2nx^2 - 2n + x^{-2n+2} - x^{-2} + 2n x^{-2} - 2n}{x^2 - 1} ]

[ = \frac{x^{2n+2} + x^{-2n+2} - x^2 - x^{-2} + 4n}{x^2 - 1} ]

[ = \frac{x^{2n+2} + x^{-2n+2} - (x^2 + x^{-2}) + 4n}{x^2 - 1} ]

[ = \frac{x^{2n+2} + x^{-2n+2} - (x^2 + \frac{1}{x^2}) + 4n}{x^2 - 1} ]

This gives the sum ( S ) in terms of ( x ) and ( n ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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