Series and Integral Test

Series

A sum of an ordered list of infinitely many real numbers.

Partial Sum (Sn)

The sum of the first n terms of a series.

Convergent Series

A series whose sequence of partial sums approaches a finite limit.

Divergent Series

A series whose partial sums do not approach a finite value.

Test for Divergence

If lim (n→∞) aₙ ≠ 0, the series ∑aₙ diverges.

Telescoping Series

A series where many terms cancel out, leaving a simpler expression to find the sum.

Geometric Series

A series in the form a + ar + ar² + ar³ + ... with common ratio r.

Convergent Geometric Series

A geometric series where |r| < 1, sum = a / (1 - r).

Divergent Geometric Series

A geometric series where |r| ≥ 1, which diverges.

Harmonic Series

The series ∑(1/n), which diverges even though the terms go to zero.

Integral Test

Used to determine convergence by comparing a series to an improper integral.

Integral Test Conditions

Function f(x) must be positive, continuous, and decreasing on [a, ∞).

Integral Test Result

If ∫ₐ^∞ f(x) dx converges, then ∑aₙ converges; if it diverges, so does ∑aₙ.

p-Series

A series of the form ∑(1/n^p).

Convergent p-Series

A p-series where p > 1.

Divergent p-Series

A p-series where p ≤ 1.

Limit of Series,The value approached by the sequence of partial sums Sn as n → ∞.

Sn (nth Partial Sum),The sum of the first n terms of a series.

Convergence Criterion for Series,If lim Sn = S (a finite number), then the series ∑aₙ converges.

Divergence Criterion for Series,If lim aₙ ≠ 0, then the series ∑aₙ diverges.

Convergent Series (lim aₙ = 0),The terms of a convergent series approach zero, but this alone does not guarantee convergence.

Harmonic Series,∑(1/n) diverges, even though its terms go to zero.

Integral Test Hypotheses,Function must be continuous, positive, and decreasing on [a, ∞).

Integral Test Conclusion,If the improper integral of f(x) converges, the related series ∑aₙ converges, and vice versa.

Improper Integral,A definite integral where one or both limits are infinite or the integrand becomes infinite.

p-Test for Series,A shortcut for ∑(1/n^p): converges if p > 1, diverges if p ≤ 1.

Telescoping Series Evaluation,Write as a difference of fractions and observe cancellation in the partial sums.

Limit Comparison,Compare a series to a known convergent or divergent one using the limit of their ratio.

Geometric Series Formula,If |r| < 1, ∑arⁿ = a / (1 - r); otherwise, it diverges.

False Convergence Assumption,Just because lim aₙ = 0, the series may still diverge.

Exponential Convergence Behavior,Terms like e^(−n²) shrink fast; series often converge.

Logarithmic Series Behavior,Series like ∑1/(n ln(n)) often diverge; need Integral Test to confirm.