Second Symmetric Sum

Algebra Level 4

\[ \displaystyle \lim_{n \to \infty } \frac {S_1 \ S_n + S_2 \ S_{n-1} + S_3 \ S_{n-2} + S_4 \ S_{n-3} + \ldots + S_n S_1 }{S_1 ^2 + S_2 ^2+S_3 ^2 + \ldots S_n ^2} \]

For natural numbers \(n\), denote \(S_n\) as the sum of an infinite geometric series whose first term is \(n\) and common ratio of \( \frac 1 {n+1} \)

If the limit above equals to \(X\), what is the value of \(10 \times X \)?

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