Final answer:
The question asks for mathematical operations on a series, including sum, average, integration, and derivation, as well as statistical analysis. Without the specific series or context, general procedures are outlined for these operations.
Step-by-step explanation:
The original question appears to involve mathematical series and statistics, including concepts such as finding a sum, average, integration, and derivation. Without the specific series or additional context, we can't provide exact methods or answers. However, we can outline general steps for each part:
- Finding the sum of a series typically involves identifying if the series is arithmetic, geometric, or follows another rule and applying the appropriate formulas or methods.
- To find the average (also known as the mean) of a series, you would sum all terms and divide by the number of terms.
- Integrating a series involves finding the antiderivative or area under the curve if the series represents a function over a continuous interval.
- To take the derivative of a series that represents a function, you would differentiate the function term-by-term if applicable.
For statistical analysis, the sample mean and sample standard deviation are calculated using statistical formulas that consider each data point in the series. To compare these to the theoretical mean and standard deviation, one would look at assumptions or models that the data is expected to follow.