Convergence and Divergence of Series

Introduction

• Review of different tests to determine if a series converges or diverges.
• Key tests discussed include: Divergence Test, Geometric Series, P-Series Test, Telescoping Series, Integral Test, Ratio Test, Root Test, Direct Comparison Test, Limit Comparison Test, Alternating Series Test.

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; further testing needed.

Geometric Series

• Identify form: constant ( a ), common ratio ( r ) raised to ( n ) or ( n-1 ).
• ( |r| < 1 ): series converges.
• ( |r| \geq 1 ): series diverges.

P-Series Test

• Form: ( 1/n^p ).
• ( p > 1 ): series converges.
• ( p \leq 1 ): series diverges.

Telescoping Series

• Write terms such that middle terms cancel.
• Evaluate partial sum formula as ( n \to \infty ).
• Finite value ( L ): series converges; otherwise, diverges.

Integral Test

• ( f(x) ): positive, continuous, decreasing function.
• Take integral from 1 to ( \infty ) of ( f(x) , dx ).
• Finite value: series converges; ( \pm \infty ) or undefined: diverges.

Ratio Test

• Limit as ( n \to \infty ) of ( |a_{n+1}/a_n| ).
• Result < 1: converges; > 1 or ( +\infty ): diverges.
• Limit = 1: inconclusive.

Root Test

• Limit as ( n \to \infty ) of ( n )-th root of ( |a_n| ).
• Value < 1: converges; > 1 or infinity: diverges.
• Value = 1: inconclusive.

Direct Comparison Test

• Two sequences ( a_n ) and ( b_n ).
• If big series converges, small series converges.
• If small series diverges, big series diverges.

Limit Comparison Test

• Limit as ( n \to \infty ) of ( a_n/b_n ).
• Equal to finite positive number ( L ): both series converge or diverge.

Alternating Series Test

• Form: ( (-1)^n \times a_n ).
• Conditions: Limit ( n \to \infty ) of ( a_n = 0 ); sequence is decreasing.
• Absolute Convergence: if ( |\text{series}| ) converges, original series converges.
• Conditional Convergence: original series converges, but ( |\text{series}| ) diverges.

Examples and Applications

• Used discussed tests to solve specific problems involving series:
  • Divergence Test application using L'Hopital's Rule.
  • Geometric Series convergence testing and sum calculation.
  • Alternating Series condition check.
  • Direct and Limit Comparison Tests for convergence of particular series.
  • Integral Test verification with improper integrals.
  • Root and Ratio Tests for convergence analysis.