Exam Problem-Solving Strategies

1. Series Convergence/Divergence

Key Strategies:

• Ratio Test: Most common for power series. Compute limit of |a_{n+1}/a_n|.
  • Series converges if limit < 1, diverges if limit > 1.
  • Indeterminate forms (0/0 or ∞/∞) use L'Hôpital's rule.
• Alternating Series Test: For series with alternating signs.
  • Terms must decrease in absolute value and approach 0.
• Comparison Tests: Compare with known convergent/divergent series.
  • Use p-series (p > 1 converges) or geometric series.
• Integral Test: When terms are positive and decreasing.
  • Evaluate ∫f(x)dx from 1 to ∞; if converges, series converges.
• Absolute Convergence: Check if |a_n| converges (implies conditional convergence).

Example:
For ∑ (-1)^n / n^2:

• Alternating series test: terms decrease to 0.
• |a_n| = 1/n^2, p-series with p=2 converges → absolutely convergent.

2. Surface Integrals & Vector Calculus

Key Strategies:

• Divergence Theorem: For closed surfaces.
  • Flux = ∫∫_S F·dS = ∫∫∫_E div F dV.
  • Ensure proper orientation (outward normal).
• Stokes' Theorem: For surface integrals of curls.
  • ∫∫_S curl F·dS = ∫_C F·dr.
  • Use parametrization for surfaces or curves.
• Parametrization: Common parametrizations:
  • Cylinders: x^2 + y^2 = r^2 → cylindrical coordinates.
  • Paraboloids: z = x^2 + y^2 → polar coordinates.
• Orientation:
  • Right-hand rule for curves/surfaces.
  • Inward/outward normals specified.

Example:
For surface z = x^2 + y^2, parametrize as r(x,y) = ⟨x, y, x^2 + y^2⟩.

• Compute ∂r/∂x × ∂r/∂y.
• Apply divergence/Stokes' theorem as needed.

3. Taylor/Maclaurin Series

Key Strategies:

• Geometric Series: Recognize patterns like 1/(1-x) = ∑x^n.
• Binomial Expansion: (1 + x)^a = ∑ C(a, n) x^n.
• Derivatives: Use f^{(n)}(0) for Maclaurin series.
• Term-by-Term Integration/Differentiation: Valid within radius of convergence.
• Error Bounds:
  • Alternating series: |R_N| ≤ |a_{N+1}|.
  • Taylor remainder: R_N = f^{(n+1)}(c)/ (n+1)!

Example:
For f(x) = e^x, Maclaurin series ∑ x^n / n!.

• Radius of convergence: all x (|x| < ∞).

4. Error Bounds & Approximations

Key Strategies:

• Alternating Series Error: |S - S_N| ≤ |a_{N+1}|.
• Integral Test Error: For decreasing f, |R_N| ≤ ∫_N^∞ f(x) dx.
• Lagrange Remainder: For Taylor series, R_N = f^{(n+1)}(c) / (n+1)! * (x-a)^{n+1}.

Example:
For sin(x) ≈ x - x^3/6 + x^5/120 (n=5), |error| ≤ |sin^{(6)}(c)| / 720 * |x|^6 ≤ 1/720 * |x|^6.

5. Repeating Decimals

Key Strategies:

• Express as geometric series.
• Let x = 0.abcdef... (repeating part).
• Multiply by 10^k (k = length of repeating part).
• Solve equation x = 10^k x - integer part.

Example:
For 0.142857142857... (repeating every 6 digits):

• Let x = 0.142857142857...
• 1000000x = 142857.142857...
• 1000000x - x = 142857 → x = 142857 / 999999.

6. Common Problem Types

a. Interval of Convergence

• Use ratio test for power series ∑ a_n (x-a)^n.
• Check endpoints separately.

b. Surface Integral Setup

• Identify surface type (disk, cylinder, sphere).
• Use appropriate parametrization.
• Compute normal vector (cross product of partial derivatives).

c. Flux Through Surfaces

• Apply divergence/Stokes' theorem.
• Ensure correct normal orientation.

d. Series Summation

• Recognize telescoping series.
• Use properties of known series (e.g., sin, cos expansions).

Tips for Success

1. Identify the type of problem first (series, integral, vector calculus).
2. Check convergence before summing.
3. Verify orientation for surface integrals.
4. Use series expansions for limits and approximations.
5. Practice parametrization of common surfaces.
6. Know your tests (Ratio, Alternating, Comparison, Integral).
7. Double-check error bounds for approximations.