Calculus II Series Tests

Flashcards for Calculus II Series Tests

Telescoping Series

A series where most terms cancel.

• Conditions: Express a_n as f(n) - f(n+1). Rewrite series using partial fractions.
• Best Use: When terms in the series can be expressed as a difference.
• How to Use: Terms cancel in the sum, take the limit of the partial sum.
• Result: Determines convergence or divergence.

Divergence Test

• Name: Divergence Test
• Conditions: lim (n→∞) a_n != 0
• Best Use: Use first to quickly check if a series diverges
• How to Use: If the limit of a_n is not 0 or DNE, the series diverges.
• Result: Only proves divergence; cannot prove convergence.

Integral Test

• Name: Integral Test
• Conditions: f(x) is positive, continuous, and decreasing
• Best Use: Use when a_n is similar to a function that can be easily integrated.
• How to Use: Evaluate the integral of a_n; if convergent, the series converges; if divergent, the series diverges.
• Result: Determines convergence or divergence.

P-Series Convergence

• Name: P-Series Test
• Conditions: Series of the form ∑ 1/n^p, f(x) is positive, continuous, and decreasing.
• Best Use: Use when the given series matches the form of a p-series.
• How to Use: Series converges if p > 1 and diverges if p <= 1.
• Result: Determines convergence or divergence.

Direct Comparison Test

• Name: Direct Comparison Test (DCT)
• Conditions: Compare an to a known convergent/divergent series bn. 0 ≤ an ≤ bn or an ≥ bn ≥ 0
• Best Use: When the series resembles a geometric or p-series.
• How to Use: If 0 ≤ an ≤ bn and ∑bn converges, then ∑an converges. If an ≥ bn ≥ 0 and ∑bn diverges, then ∑an diverges.
• Result: Determines convergence or divergence.

Alternating Series

A series where the signs alternate.

Absolute/Conditional Convergence

If the absolute value of a_n converges, it is absolutely convergent; otherwise, it is conditionally convergent.

Geometric Series Test

• Name: Geometric Series Test
• Conditions: Series of the form ∑ ar^n
• Best Use: Use when the series has a constant ratio between terms.
• How to Use: If |r| < 1, the series converges to a/(1-r). If |r| >= 1, it diverges.
• Result: Determines convergence or divergence.

P-Series Test

• Name: P-Series Test
• Conditions: Series is of the form ∑1/n^p
• Best Use: Use when the series matches the form of a p-series.
• How to Use: If p > 1, the series converges. If p ≤ 1, it diverges.
• Result: Determines convergence or divergence.

Integral Test Conditions

Function f(n) = a_n is positive, continuous, decreasing. If the integral converges, the series converges. If the integral diverges, so does the series.

Direct Comparison Test (DCT)

Compare an to a known convergent or divergent series bn. If 0 ≤ an ≤ bn and ∑bn converges, then ∑an converges. If an ≥ bn ≥ 0 and ∑bn diverges, then ∑an diverges.

Limit Comparison Test (LCT)

• Name: Limit Comparison Test (LCT)
• Conditions: Both series must be positive. Compute L = lim (n→∞) an/bn
• Best Use: When you suspect the series behaves similarly to another series but direct comparison is difficult.
• How to Use: Compute L = lim (n→∞) an/bn. If 0 < L < ∞, then both series converge or diverge together.
• Result: Determines convergence or divergence for both tested series.

Alternating Series Test (AST or Leibniz)

• Name: Alternating Series Test (AST or Leibniz)
• Conditions: Series has the form ∑(-1)^n bn. Confirm bn is decreasing and lim (n→∞) b_n = 0
• Best Use: Use for alternating series.
• How to Use: Verify that bn is decreasing and lim (n→∞) bn = 0. If true, the series converges.
• Result: Determines convergence (possibly conditionally).

Absolute/Conditional Convergence Test

• Name: Absolute/Conditional Convergence Test
• Conditions: Consider ∑|a_n|
• Best Use: After determining that a series converges using the Alternating Series Test.
• How to Use: If ∑|an| converges, the series is absolutely convergent. If it diverges, but original ∑an converges → conditionally convergent.
• Result: Determines absolute or conditional convergence.

Ratio Test

• Name: Ratio Test
• Conditions: None
• Best Use: Use when series involves factorials or exponential terms.
• How to Use: Compute L = lim (n→∞) |(a{n+1})/an|
• Result:
  • If L < 1: converges absolutely.
  • If L > 1 or ∞: diverges.
  • If L = 1: inconclusive.

Root Test

• Name: Root Test
• Conditions: None
• Best Use: Use when series involves terms raised to the nth power.
• How to Use: Compute L = lim (n→∞) n^{th} root(|a_n|).
• Result:
  • If L < 1: converges absolutely.
  • If L > 1: diverges.
  • If L = 1: inconclusive.

Telescoping Series Procedure

Write a_n = f(n) - f(n+1). Rewrite the series in terms of partial fractions. Terms cancel in the sum, take the limit of the partial sum.

Harmonic Series

• Name: Harmonic Series
• Conditions: Series of the form ∑1/n
• Best Use: As a common series for comparison.
• How to Use: Recognize that ∑1/n diverges.
• Result: Diverges, even though a_n → 0.