Study Guide for Series and Convergence Tests

1. Understanding Sequences and Series

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• Sequence: An ordered list of numbers, typically defined by a formula or recurrence relation.

• Series: The sum of the terms of a sequence.

• Partial Sums: The sum of the first nnn terms, used to analyze convergence.

2. Integral Test and Its Application

• Used when a function f(x)f(x)f(x) is positive, continuous, and decreasing.

• Helps determine if a corresponding series behaves like an improper integral.

• Requires setting up an integral and evaluating its behavior.

Common Functions Used in Integral Test

• Power functions: 1xp\frac{1}{x^p}xp1​

• Logarithmic functions: 1xln⁡x\frac{1}{x \ln x}xlnx1​

3. Surface Area of a Revolution

• The formula for rotating a function y=f(x)y = f(x)y=f(x) about the x-axis: A=∫ab2πy1+(dydx)2dxA = \int_{a}^{b} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dxA=∫ab​2πy1+(dxdy​)2​dx

• Key skills:

  • Computing derivatives of functions.

  • Setting up definite integrals over given bounds.

4. Recurrence Relations and Closed-Form Sequences

• A recurrence relation expresses a term using previous terms.

  • Example: an=r⋅an−1a_n = r \cdot a_{n-1}an​=r⋅an−1​

• A closed-form formula provides an explicit expression for any term ana_nan​ without recursion.

• Important in analyzing geometric sequences and exponential growth.

5. Monotone Convergence and Alternating Series

• A sequence is monotonic if it is either increasing or decreasing.

• Some tests rely on checking whether a sequence is bounded and monotonic.

• Alternating series involve terms that switch signs, and their behavior depends on term magnitude.

6. Comparing Series for Convergence

Direct Comparison

• Compares a given series with a known reference series.

• Used when the terms of one series are larger or smaller than another known series.

Limit Comparison

• Uses the limit of the ratio of two sequences to compare their behavior.

• Requires evaluating the limiting behavior of their terms.

7. Absolute vs. Conditional Convergence

• Absolute convergence: A series ∑an\sum a_n∑an​ converges absolutely if ∑∣an∣\sum |a_n|∑∣an​∣ also converges.

• Conditional convergence: A series converges but does not converge absolutely.

8. Ratio and Root Tests

• Both tests are useful for series involving factorials or exponentials.

• They analyze the growth rate of terms using ratios or roots.

When to Use These Tests

• Ratio Test is effective for factorial terms (e.g., n!n!n!) or exponential terms (e.g., rnr^nrn).

• Root Test is useful when terms involve nth powers.

9. Geometric Series and Solving for rrr

• A geometric series has the form: ∑arn\sum ar^n∑arn

• Requires solving for r when given a sum equation.

10. Integral Evaluation for Series Tests

• Some tests require computing improper integrals to analyze convergence.

• Example: Converting a sum into an integral approximation.

11. Strategies for Test Selection

• If terms have factorials or exponentials: Try the Ratio or Root Test.

• If comparing to a known series: Use the Comparison or Limit Comparison Test.

• If terms alternate in sign: Check for the Alternating Series Test.

• If terms are defined by an integral function: Consider the Integral Test.