Find the sum to n terms of the series: `(x+1/x)^(2)+(x^(2)+1/(x^(2)))^(2)+(x^(3)+1/(x^(3)))^(2)`+`……

Answer

To find the sum to n terms of the series:

\[ \left(x + \frac{1}{x}\right)^2 + \left(x^2 + \frac{1}{x^2}\right)^2 + \left(x^3 + \frac{1}{x^3}\right)^2 + \ldots + \left(x^n + \frac{1}{x^n}\right)^2 \]

we will follow these steps:

### Step 1: Expand Each Term
Each term in the series can be expanded using the identity \((a + b)^2 = a^2 + b^2 + 2ab\).

For the k-th term, we have:
\[ \left(x^k + \frac{1}{x^k}\right)^2 = \left(x^k\right)^2 + \left(\frac{1}{x^k}\right)^2 + 2\left(x^k \cdot \frac{1}{x^k}\right) = x^{2k} + \frac{1}{x^{2k}} + 2 \]

Thus, the k-th term simplifies to:
\[ x^{2k} + \frac{1}{x^{2k}} + 2 \]

### Step 2: Write the Series
Now, we can write the sum of the first n terms:
\[ S_n = \sum_{k=1}^{n} \left(x^{2k} + \frac{1}{x^{2k}} + 2\right) \]

This can be separated into three sums:
\[ S_n = \sum_{k=1}^{n} x^{2k} + \sum_{k=1}^{n} \frac{1}{x^{2k}} + \sum_{k=1}^{n} 2 \]

### Step 3: Calculate Each Sum
1. **Sum of \(x^{2k}\)**:
This is a geometric series with first term \(x^2\) and common ratio \(x^2\):
\[ \sum_{k=1}^{n} x^{2k} = x^2 \frac{(x^2)^n - 1}{x^2 - 1} = \frac{x^2(x^{2n} - 1)}{x^2 - 1} \]

2. **Sum of \(\frac{1}{x^{2k}}\)**:
This is also a geometric series with first term \(\frac{1}{x^2}\) and common ratio \(\frac{1}{x^2}\):
\[ \sum_{k=1}^{n} \frac{1}{x^{2k}} = \frac{\frac{1}{x^2} \left(\left(\frac{1}{x^2}\right)^n - 1\right)}{\frac{1}{x^2} - 1} = \frac{\frac{1}{x^2}(1 - \frac{1}{x^{2n}})}{1 - \frac{1}{x^2}} = \frac{1 - \frac{1}{x^{2n}}}{x^2 - 1} \]

3. **Sum of the constant \(2\)**:
\[ \sum_{k=1}^{n} 2 = 2n \]

### Step 4: Combine the Results
Now, we combine all three sums:
\[ S_n = \frac{x^2(x^{2n} - 1)}{x^2 - 1} + \frac{1 - \frac{1}{x^{2n}}}{x^2 - 1} + 2n \]

### Step 5: Simplify
Combine the fractions:
\[ S_n = \frac{x^2(x^{2n} - 1) + 1 - \frac{1}{x^{2n}}}{x^2 - 1} + 2n \]

### Final Result
Thus, the sum to n terms of the series is:
\[ S_n = \frac{x^2(x^{2n} - 1) + 1 - \frac{1}{x^{2n}}}{x^2 - 1} + 2n \]