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 $\lim<em>{n \to \infty} a</em>n \neq 0$ or if $\lim<em>{n \to \infty} a</em>n$ does not exist.
  • Inconclusive if $\lim<em>{n \to \infty} a</em>n = 0$.

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 $\sum<em>{n=1}^ ext{∞} b</em>n$ converges and $a<em>n \leq b</em>n$ for all n, then $\sum<em>{n=1}^ ext{∞} a</em>n$ converges.
• Conditions for Divergence:
  • If $\sum<em>{n=1}^ ext{∞} b</em>n$ diverges and $a<em>n \geq b</em>n$ for all n, then $\sum<em>{n=1}^ ext{∞} a</em>n$ 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 $a<em>n > 0$ and bn > 0.
• Criteria:
  • If $\lim<em>{n \to \infty} \frac{a</em>n}{b_n} = c$ 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. $b<em>{n+1} \leq b</em>n$ for all n (i.e., the sequence ${b_n}$ is decreasing).
  2. $\lim<em>{n \to \infty} b</em>n = 0$.

Absolute Convergence

• Definition: extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} an \bigg)
• Convergence:
  • Converges absolutely (and therefore converges) if $\sum<em>{n=1}^\infty |a</em>n|$ 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 $\sum<em>{n=1}^\infty a</em>n$ converges but $\sum<em>{n=1}^\infty |a</em>n|$ diverges.

Ratio Test

• Definition: extstyleigg( \, extstyle igg(
  otag\sum{n=1}^ ext{∞} an \bigg)
• Criteria:
  • If $\lim<em>{n \to \infty} \left| \frac{a</em>{n+1}}{a_n} \right| = L$, 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: $\sum<em>{n=1}^ ext{∞} a</em>n$
• Criteria:
  • If $\lim<em>{n \to \infty} \sqrt[n]{|a</em>n|} = L$, 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.