Convergence and Divergence of Series

Divergence Test

• Take limit as ( n \to \infty ) of sequence ( a_n ).
• If limit ( \neq 0 ), series diverges.
• If limit = 0, series may converge or diverge (use another test).

Geometric Series

• Form: ( a r^n ) or ( a r^{n-1} ).
• Identify ( r ):
  • If ( |r| < 1 ), series converges.
  • If ( |r| \geq 1 ), series diverges.

P-Series Test

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

Telescoping Series

• Form where terms cancel in sequence (e.g., ( 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} )).
• Write partial sum formula.
• Evaluate sum as ( n \to \infty ).
  • Finite value: converges.
  • ( \pm \infty ) or DNE: diverges.

Integral Test

• Convert sequence ( a_n ) to function ( f(x) ).
• ( f(x) ) must be positive, continuous, decreasing from some ( n \to \infty ).
• Evaluate integral from 1 to infinity:
  • Finite value: converges.
  • ( \pm \infty ) or doesn't exist: diverges.

Ratio Test

• Take limit as ( n \to \infty ) of ( \left| \frac{a_{n+1}}{a_n} \right| ).
• If ( < 1 ), converges.
• If ( > 1 ) or ( \pm \infty ), diverges.
• If = 1, inconclusive.

Root Test

• Take limit as ( n \to \infty ) of ( \sqrt[n]{|a_n|} ).
• If ( < 1 ), converges.
• If ( > 1 ) or ( \infty ), diverges.
• If = 1, inconclusive.

Direct Comparison Test

• Compare two sequences ( a_n ) and ( b_n ).
• If large series converges, small series converges.
• If small series diverges, large series diverges.

Limit Comparison Test

• Compare two sequences ( a_n ) and ( b_n ).
• Take limit as ( n \to \infty ) of ( \frac{a_n}{b_n} ).
  • If finite positive value ( L ), both series converge or diverge.

Alternating Series Test

• Series with alternating signs.
• Conditions:
  1. Pass divergence test (limit to 0).
  2. Sequence ( a_n ) decreases.
• Absolutely convergent if absolute value series converges.
• Conditionally convergent if absolute value series diverges but original converges.

Example Problems

• Divergence Test Example:

  • Series ( \sum \frac{2n^2 + 5}{7n^2 - 4} ), diverges as limit ( \neq 0 ).

• P-Series Example:

  • Series ( \sum \frac{\sqrt[3]{n}}{n^5} ) reduces to ( \frac{1}{n^{14/3}} ) converges as ( p > 1 ).

• Geometric Series Example:

  • ( 5 \times \left(\frac{1}{4}\right)^{n-1} ), converges (( r < 1 )).
  • Sum = 20/3.

• Alternating Series Example:

  • ( \sum \frac{(-1)^n}{\sqrt{n}} ):
    • Passes divergence test.
    • Sequence decreases.
    • Conditionally convergent (absolute value diverges).

• Telescoping Series Example:

  • ( \sum \frac{1}{n(n+1)} ):
    • Partial fraction decomposition to ( \frac{1}{n} - \frac{1}{n+1} ).
    • Sum = 1 (converges).

• Integral Test Example:

  • Series ( \sum \frac{1}{\sqrt{n} - 2} ), diverges, integral test shows infinity.

• Limit Comparison Test Example:

  • Series ( \frac{\sqrt{n}}{n^3 + 2} ), converges (comparison to ( \frac{1}{n^{2.5}} )).

• Root Test Example:

  • ( \sum \left(\frac{3n^2 - 9}{7n^2 + 4}\right)^n ) converges as limit = ( \frac{3}{7} < 1 ).

• Ratio Test Example:

  • ( \sum \frac{2^n}{n!} ) converges, ratio test shows limit = 0.