Series Convergence and Divergence Tests

These flashcards cover key concepts related to the methods of determining convergence or divergence of series.

Divergence Test

If the limit of a_n as n approaches infinity does not equal zero, the series diverges.

Geometric Series

A series in the form a * r^n or a * r^(n-1) with a constant a and common ratio r; diverges if |r| >= 1.

p Series Test

If the series is in the form 1/n^p, it converges if p > 1 and diverges if p ≤ 1.

Telescoping Series

A series where intermediate terms cancel out; requires finding the general formula for the partial sum.

Integral Test

If the integral from 1 to infinity of f(x) converges to a finite value, then the series converges; otherwise, it diverges.

Ratio Test

If lim (n→∞) |a(n+1) / an| < 1, the series converges; if > 1, it diverges; if = 1, inconclusive.

Root Test

If lim (n→∞) (|a_n|)^(1/n) < 1, the series converges; if > 1, it diverges; if = 1, inconclusive.

Direct Comparison Test

If an is less than bn and bn converges, then an converges; if an is greater than bn and an diverges, then bn diverges.

Limit Comparison Test

If lim (n→∞) (an / bn) = c (finite, positive), both series either converge or diverge.

Alternating Series Test

For a series with alternating signs to converge, it must pass the divergence test (limit is zero) and a_n must be decreasing.

Absolute Convergence

If the absolute value of the series converges, then the original series also converges.

Conditional Convergence

If the series converges, but the absolute value diverges, then it is conditionally convergent.

Divergent Series

If both the original series and its absolute value diverge, the series is divergent.