It appears there's a bit of confusion in the formula of the series. However, I'll try my best to provide a detailed explanation based on the possible interpretation of the series.
The given series seems to be a mixture of a geometric and power series with the following format:
∑ [n=0 -> ∞] 2^(-n) * (5+sinx)^n / {2*(6+sinx)*(2*(6-sinx)*5*(6+sinx)*2*(6-sinx)}.
The general form of a geometric series is ∑ [n=0 -> ∞] a*r^n, where a is the first term and r is the common ratio. If -1 < r < 1, the series converges and its sum can be given by a / (1 - r).
In this case, the first term 'a' is 1 (when n = 0) and the common ratio 'r' can be interpreted as 2^(-n) * (5+sinx)^n.
A geometric series converges if and only if the absolute value of the common ratio is less than 1. In mathematical terms, it means |r| < 1.
Therefore, the task is:
1) To find the range of 'x' for which the absolute value "|2^(-n) * (5+sinx)^n|" is less than 1. This can be achieved by setting up an inequality and solving for 'x'.
2) If this condition is met and the series converges, calculate the sum of the series using the formula a / (1 - r), where a = 1 and r = 2^(-n) * (5+sinx)^n.
However, due to the complex nature of this series, a closed form solution might not be possible or could be beyond traditional mathematical methods. Alternatively, numerical solutions can be employed to approximate both the range of convergence and the sum of this series. It's important to note that these calculations should be done carefully as errors can propagate and lead to incorrect results.