Jacobsthal-Geometric Series

Algebra Level 5

\[\text{J}_{n}=\begin{cases} 0 \quad & \text{if}~n=0; \\ 1 \quad & \text{if} ~ n=1; \\ \text{J}_{n-1}+2\text{J}_{n-2} \quad & \text{if}~n>1.\end{cases} \]

Jacobsthal Numbers are defined by the recurrence relation as described above. Evaluate: \[\displaystyle \sum_{n=0}^{\infty} \dfrac{\text{J}_{n}}{10^{n+1}} \; .\]

If the value of the above expression is in the form \(A^{-1}\), find \(A\).


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