The document contains a range of mathematical problems from calculus, linear algebra, and series convergence tests. Below are the steps I recommend to address these:

1. Choose specific problems: Let me know which problems you would like to solve first.
2. Detail and clarification: If you have specific doubts about the questions or methods, feel free to ask.

Would you like me to solve all the problems in sequence, or do you want to start with specific sections?

Meanwhile, here’s a small breakdown of the questions that can be solved:

1. Taylor expansions for polynomials and trigonometric functions.
2. Multivariable calculus questions (gradients, divergence, curl).
3. Linear algebra (Cayley-Hamilton theorem, matrix inverses, eigenvalues).
4. Series convergence tests using different methods.
5. Curvature and surface angles at specific points.

Let me know how you’d like to proceed!

Here are 5 related questions to expand your understanding:

1. What is Taylor's theorem, and how is it applied to approximate functions?
2. How do you find the divergence and curl of a vector field?
3. Why is Cayley-Hamilton theorem important in finding matrix inverses?
4. What conditions determine convergence or divergence in infinite series?
5. How do we calculate the radius of curvature for curves in implicit forms?

Tip: When solving problems step-by-step, ensure you understand the foundational theorem or formula behind each step. This builds intuition for more complex problems.