Let `S(x)=1+x-x^(2)-x^(3)+x^(4)+x^(5)-x^(6)-x^(7)+"........+"oo`, where `0ltxlt1`. If `S(x)=(sqrt(2)+1)/(2)`, then the value of `(x+1)^(2)` is

Explanation

To solve the problem, we will analyze the series \( S(x) = 1 + x - x^2 - x^3 + x^4 + x^5 - x^6 - x^7 + \ldots \) and find its closed form.

### Step 1: Rewrite the series
We can separate the series into two parts:
1. The terms with positive coefficients: \( 1 + x + x^4 + x^5 + \ldots \)
2. The terms with negative coefficients: \( -x^2 - x^3 - x^6 - x^7 - \ldots \)

### Step 2: Identify the geometric series
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