The document contains various mathematical questions across different topics, including series, integration (definite and indefinite), approximation methods (e.g., Riemann sums, Simpson's rule), and techniques like integration by parts. Here's a brief breakdown:

Question Highlights:

1. Series and Summations:

   • Express a sum as a function of $n$.
   • Evaluate infinite series and calculate specific sums.
   • Geometric and arithmetic series evaluations.

2. Definite Integrals:

   • Calculate integrals using Riemann sums.
   • Fundamental theorem of calculus applications.

3. Approximation Techniques:

   • Simpson's rule, Trapezoidal rule, Midpoint rule, and their errors.

4. Integration by Parts:

   • Evaluate complex integrals and definite integrals using integration by parts.

5. Miscellaneous Integrals:

   • Writing and evaluating basic forms of integrals like $\int x^n \, dx$, $\int e^x \, dx$, and trigonometric integrals.

Would you like a solution to any specific part, or an explanation of a topic like Riemann sums, series summations, or integration techniques? Let me know how I can assist!