Convergence and Divergence Tests for Series

Geometric Series

• Definition: ext{Series: } \, extstyleigg( \, extstyle igg(
  otag\sum_{n=1}^ ext{∞} a r^{n-1} \, , \, a \neq 0 \bigg)
• Convergence:
  • Converges if |r| < 1.
  • Sum of the series: s = \frac{a}{1 - r}
• Divergence:
  • Diverges if |r| \geq 1.
• Utility: Useful for comparison tests if the nth term a_n is similar to ar^{n-1}.

Test for Divergence

• Definition: extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} an \bigg)
• Divergence Criteria:
  • Diverges if \lim{n \to \infty} an \neq 0 or if \lim{n \to \infty} an does not exist.
  • Inconclusive if \lim{n \to \infty} an = 0.

p-Series

• Definition: extstyleigg( \, extstyle igg(
  otag\sum_{n=1}^ ext{∞} \frac{1}{n^p} \bigg)
• Convergence:
  • Converges if p > 1.
• Divergence:
  • Diverges if p \leq 1.
• Utility: Useful for comparison tests when the nth term a_n is similar to \frac{1}{n^p}.

Integral Test

• Definition: For a series extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} an , \, a_n = f(n), \, n \geq 1 \bigg)
• Convergence:
  • Converges if \int_{1}^{\infty} f(x) \, dx converges.
• Divergence:
  • Diverges if \int_{1}^{\infty} f(x) \, dx diverges.
• Requirements:
  • The function f must be continuous, positive, and decreasing on [1, \infty).

Comparison Test

• Given two series: extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} an \text{ and } \sum{n=1}^ ext{∞} bn \bigg)
• Conditions for Convergence:
  • If \sum{n=1}^ ext{∞} bn converges and an \leq bn for all n, then \sum{n=1}^ ext{∞} an converges.
• Conditions for Divergence:
  • If \sum{n=1}^ ext{∞} bn diverges and an \geq bn for all n, then \sum{n=1}^ ext{∞} an diverges.
• Comparison Series: Typically, the series b_n is a geometric series or a p-series.

Limit Comparison Test

• Given two series: extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} an \text{ and } \sum{n=1}^ ext{∞} bn \bigg) where both have an > 0 and bn > 0.
• Criteria:
  • If \lim{n \to \infty} \frac{an}{b_n} = c where c > 0 is a finite real number, then both series converge or diverge together.

Alternating Series Test

• For series: extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} (-1)^n bn \text{ where } b_n > 0 \bigg)
• Convergence Criteria:
  • Converges if:
  1. b{n+1} \leq bn for all n (i.e., the sequence {b_n} is decreasing).
  2. \lim{n \to \infty} bn = 0.

Absolute Convergence

• Definition: extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} an \bigg)
• Convergence:
  • Converges absolutely (and therefore converges) if \sum{n=1}^\infty |an| converges.
• Utility: Applicable for series including both positive and negative terms.

Conditional Convergence

• Definition: extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} an \bigg)
• Criteria:
  • If \sum{n=1}^\infty an converges but \sum{n=1}^\infty |an| diverges.

Ratio Test

• Definition: extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} an \bigg)
• Criteria:
  • If \lim{n \to \infty} \left| \frac{a{n+1}}{a_n} \right| = L, then:
  • The series is absolutely convergent (and therefore converges) if L < 1.
  • Divergent if L > 1 or L = \infty.
  • Inconclusive if L = 1.
• Utility: Particularly useful for terms involving factorials n! or nth powers.

Root Test

• Definition: \sum{n=1}^ ext{∞} an
• Criteria:
  • If \lim{n \to \infty} \sqrt[n]{|an|} = L, then:
  • The series is absolutely convergent (and therefore convergent) if L < 1.
  • Divergent if L > 1 or L = \infty.
  • Inconclusive if L = 1.
• Utility: Particularly useful for terms involving nth powers.

Note:

• If for all n, a_n > 0, the absolute value sign may be disregarded in ratio and root tests.