Key Concepts in Series Analysis

These notes summarize the various tests used to determine whether an infinite series converges or diverges.

1. Divergence Test

• Objective: Determine if a series diverges.
• Method: Calculate the limit of a sequence ( a_n ) as ( n \to \infty ).
  • If ( \lim_{n \to \infty} a_n \neq 0 ), the series diverges.
  • If ( \lim_{n \to \infty} a_n = 0 ), the series may converge; use another test.

2. Geometric Series Test

• Form: ( ar^n ) or ( ar^{n-1} ).
• Convergence Criteria:
  • If ( |r| < 1 ), the series converges.
  • If ( |r| \geq 1 ), the series diverges.

3. P-Series Test

• Form: ( \sum \frac{1}{n^p} ).
• Convergence:
  • If ( p > 1 ), the series converges.
  • If ( p \leq 1 ), the series diverges.

4. Telescoping Series

• Convergence: Involves canceling terms to find a partial sum.
• Method:
  • Write the general formula for partial sums and evaluate the limit.
  • If it converges to a finite value, the series converges.

5. Integral Test

• Conditions: Function ( f(n) ) is positive, continuous, and decreasing.
• Convergence:
  • Evaluate ( \int_1^\infty f(x) , dx ).
  • Finite value indicates convergence; infinity or doesn’t exist indicates divergence.

6. Ratio Test

• Method: Calculate ( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ).
  • If the result is < 1, the series converges.
  • If > 1 or infinity, it diverges.
  • If = 1, the test is inconclusive.

7. Root Test

• Method: ( \lim_{n \to \infty} \sqrt[n]{|a_n|} ).
  • If < 1, converges.
  • If > 1 or infinity, diverges.
  • If = 1, inconclusive._

8. Direct Comparison Test

• Compare two series ( a_n ) (smaller) and ( b_n ) (larger).
• Convergence:
  • ( b_n ) converges implies ( a_n ) converges.
  • ( a_n ) diverges implies ( b_n ) diverges.

9. Limit Comparison Test

• Method: Compare series using ( \lim_{n \to \infty} \frac{a_n}{b_n} = L ).
  • If ( L > 0 ), both series converge or diverge together._

10. Alternating Series Test

• Form: (-1)(^{n} a_n).
• Convergence:
  • Must pass the divergence test (limit of ( a_n \to 0 )).
  • ( a_{n+1} < a_n ) for convergence.
  • Conditional Convergence: Original converges, but absolute diverges._

11. Examples and Application of Tests

• Divergence Test: Showed divergence for ( \frac{2n^2 + 5}{7n^2 - 4} ).

• P-Series: ( \frac{n^{1/3}}{n^5} ) converges as ( p = \frac{14}{3} > 1.

• Geometric Series: ( 5 \left( \frac{1}{4} \right)^{n-1} ) converges with (|r|=\frac{1}{4} < 1).

• Alternating Series: Showed conditional convergence for ( \frac{(-1)^n}{\sqrt{n}} ).

• Telescoping Series: Analyzed ( \frac{1}{n(n+1)} ). Converges to 1 using partial fractions.

• Direct Comparison: ( \frac{1}{n^2+4} ) converges compared to ( \frac{1}{n^2} ).

• Integral Test: ( \frac{1}{\sqrt{x-2}} ) diverges using improper integrals.

• Limit Comparison: ( \frac{\sqrt{n}}{n^3+2} ) converges compared to ( \frac{\sqrt{n}}{n^3} ).

• Root Test: Indicated convergence of ( \left(\frac{3n^2-9}{7n^2+4}\right)^n ).

• Ratio Test: Convergent series ( \frac{2^n}{n!} ) with ( \lim \to 0 ).

These notes summarize key methods to evaluate the convergence or divergence of a series, useful for mathematical analysis and problem-solving in calculus.