# 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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