Sum of a Series

Calculus Level 5

Let \(\{ a_n \} \) be a sequence defined as \(a_0 = a_1 = 1 \) and \( a_{n+2} = 2 a_{n+1} + a_n - 1 \) for \( n \ge 0 \).

Find the value of this infinite summation \[S = \sum_{k=0}^{\infty} \frac{a_k}{3^k} \]