Review of Series Convergence Tests

1. Divergence Test

• Procedure: Take the limit as ( n \to \infty ) of the sequence ( a_n ).
  • If the limit ( \neq 0 ), the series diverges.
  • If the limit ( = 0 ), further testing is needed.

2. Geometric Series

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

3. P-Series Test

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

4. Telescoping Series

• Form: Cancellation of terms allows you to find a simple partial sum.
• Convergence:
  • Evaluate the partial sum formula as ( n \to \infty ).

5. Integral Test

• Conditions: Function ( f(x) ) must be positive, continuous, and decreasing.
• Procedure:
  • Take ( \int_1^\infty f(x) , dx ).
  • Finite integral implies convergence; infinite implies divergence.

6. Ratio Test

• Procedure: ( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| )
  • (< 1): Series converges.
  • (> 1) or (= \infty): Series diverges.
  • (= 1): Inconclusive.

7. Root Test

• Procedure: ( \lim_{n \to \infty} \sqrt[n]{|a_n|} )
  • (< 1): Series converges.
  • (> 1) or (= \infty): Series diverges.
  • (= 1): Inconclusive._

8. Direct Comparison Test

• Concept: Compare with a known series ( b_n ).
  • If ( b_n ) converges and ( b_n ) ( \ge a_n ), then ( a_n ) converges.
  • If ( a_n ) diverges and ( b_n \le a_n ), then ( b_n ) diverges.

9. Limit Comparison Test

• Procedure: ( \lim_{n \to \infty} \frac{a_n}{b_n} = L )
  • If ( L ) is finite and positive, both series ( a_n ) and ( b_n ) either converge or diverge together._

10. Alternating Series Test

• Form: ( (-1)^n a_n ) or ( (-1)^{n+1} a_n ).
• Conditions:
  • ( a_n \to 0 ) as ( n \to \infty ).
  • Sequence ( a_n ) is decreasing.

11. Absolute Convergence

• Concept:
  • If ( |a_n| ) converges, then the series is absolutely convergent.
  • If ( a_n ) converges but ( |a_n| ) does not, it is conditionally convergent.

Examples and Applications

Example 1

• Series: ( \frac{2n^2+5}{7n^2-4} )
• Test: Divergence Test
  • Limit ( \neq 0 ), series diverges.

Example 2

• Series: ( \frac{\sqrt[3]{n}}{n^5} )
• Test: P-Series
  • Simplified to ( \frac{1}{n^{\frac{14}{3}}} ), ( p > 1 ), series converges.

Example 3

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

Example 4

• Series: ( \frac{(-1)^n}{\sqrt{n}} )
• Test: Alternating Series
  • Meets criteria, converges conditionally.

Example 5

• Series: ( \frac{1}{n(n+1)} )
• Test: Telescoping Series
  • Converges, sum is 1.

Example 6

• Series: ( \frac{1}{n^2+4} )
• Test: Direct Comparison
  • Compared with ( \frac{1}{n^2} ), converges.

Example 7

• Series: ( \frac{1}{\sqrt{n}-2} )
• Test: Direct Comparison and Integral Test
  • Diverges.

Example 8

• Series: ( \frac{\sqrt{n}}{n^3+2} )
• Test: Limit Comparison
  • Converges using comparison with ( \frac{\sqrt{n}}{n^3} ).

Example 9

• Series: ( \left( \frac{3n^2-9}{7n^2+4} \right)^n )
• Test: Root Test
  • Limit ( < 1 ), series converges.

Example 10

• Series: ( \frac{2^n}{n!} )
• Test: Ratio Test
  • Limit goes to 0, series converges.

These summaries provide an overview of the tests necessary for series convergence or divergence, along with examples illustrating each test's application.