Find the value of \(S\) as defined below:

\[S=1+\dfrac2{10}+\dfrac3{10^2}+\dfrac{10}{10^3}+\dfrac{22}{10^4}+\dfrac{51}{10^5}+\dfrac{125}{10^6}+\dfrac{293}{10^7}+ \cdots \]

Enter 0.6666 if you come to the conclusion that the series fails to converge. Enter your Answer upto 4 decimal places

Note: The numerators follow the pattern \( T_{n+3} = T_{n+2} + 2 T_{n+1} + 3 T_n \).

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