Series Convergence Tests

These flashcards cover key vocabulary and definitions related to series convergence tests, including various essential concepts such as L'Hôpital's Rule, the Divergence Test, and specific tests for convergence and divergence.

Divergence Test

If the limit of the terms of a series does not go to zero, then the series diverges.

p-Series Test

A p-series is of the form ∑(1/k^p); it converges if p > 1 and diverges if 0 < p ≤ 1.

Limit Comparison Test

If lim (an / bn) = L, and L is positive and finite, then both series ∑an and ∑bn either converge or diverge together.

L'Hôpital's Rule

A method to evaluate limits of indeterminate forms by taking derivatives of the numerator and denominator.

Geometric Series

A series of the form ∑ar^k; converges if |r| < 1 and diverges if |r| ≥ 1.

Indeterminate Form

A limit that cannot be determined directly and requires further analysis, such as infinity/infinity or 0/0.

Limit of a Rational Function

If a function is in the form p(k)/q(k), the limit depends on the degrees of p and q (m < n, m = n, or m > n).

Comparison Test

For two series with positive terms, if ak ≤ bk and ∑bk converges, then ∑ak converges; if bk ≤ ak and ∑bk diverges, then ∑ak diverges.

Exponent Limit

In the limit lim (5k)^(1/k), the exponent approaches 0, leading to an overall limit of 1.

Condition for Divergence

If lim ak ≠ 0 for a series ∑ak, the series diverges according to the Divergence Test.