Let `S_(n),n=1,2,3,"…"` be the sum of infinite geometric series, whose first term is n and the common ratio is `(1)/(n+1)`. Evaluate `lim_(n to oo)(S_(1)S_(n)+S_(2)S_(n-1)+S_(3)S_(n-2)+"..."+S_(n)S_(1))/(S_(1)^(2)+S_(2)^(2)+"......"+S_(n)^(2))`.

Explanation

To solve the given problem, we need to evaluate the limit:

\[ \lim_{n \to \infty} \frac{S_1 S_n + S_2 S_{n-1} + S_3 S_{n-2} + \ldots + S_n S_1}{S_1^2 + S_2^2 + \ldots + S_n^2} \]

### Step 1: Find the sum of the infinite geometric series \( S_n \)

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