Convergence Tests in Calculus

These flashcards cover key concepts related to convergence tests for series.

Direct Comparison Test

If a series converges, then it is bounded by a converging series. If a series diverges, then it is bounded by a diverging series.

Limit Comparison Test

Given two series with positive terms, if the limit of their ratio is finite and positive, both series converge or diverge together.

Root Test

For a series $ext{a}$, if $ext{lim}_{k o ext{∞}} ext{( } ext{a}_k ext{ )}^{1/k} = L$, then if $L < 1$ the series converges, if $L > 1$ it diverges, and if $L = 1$, the test is inconclusive.

Alternating Series Test

For an alternating series $ext{a}<em>{k} = (-1)^{k} ext{b}</em>{k}$, if $ext{b}<em>{k}$ is decreasing and $ext{lim}</em>{k o ext{∞}} ext{b}_{k} = 0$, then the series converges.

Geometric Series

A series of the form $ext{a} + ext{ar} + ext{ar}^{2} + …$ converges if the absolute value of the common ratio |r| < 1.

P-Series

A series of the form ext{Σ}rac{1}{k^p} converges if p > 1 and diverges if $p <br /> eq 1$.

Divergence Test

If the limit of a term of the series does not approach zero, then the series diverges.

Integral Test

If ext{an} > 0, $ext{decreasing}$, and $ext{continuous}$, then the convergence of the series $ext{Σ a_n}$ can be determined by the convergence of the integral $ext{∫ a_n dx}$.