A series converges when its sequence of partial sums converges. Often the sum cannot be found directly, so convergence tests determine whether a finite sum exists without calculating it.

Absolute and conditional convergence

A series

$$\sum a_n$$

converges absolutely when

$$\sum |a_n|$$

converges. Absolute convergence guarantees convergence of the original series.

A series is conditionally convergent when $\sum a_n$ converges but $\sum|a_n|$ does not.

Comparison tests

For nonnegative terms, comparison with a known series can decide convergence. If

$$0\le a_n\le b_n$$

and $\sum b_n$ converges, then $\sum a_n$ also converges. Conversely, if $a_n\ge b_n\ge0$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

The limit comparison test compares the long-term size of two positive sequences through

$$\lim_{n\to\infty}\frac{a_n}{b_n}=L,$$

with $0<L<\infty$.

Ratio and root tests

The ratio test studies

$$L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|,$$

while the root test studies

$$L=\lim_{n\to\infty}\sqrt[n]{|a_n|}.$$

When $L<1$, the series converges absolutely; when $L>1$, it diverges. At $L=1$, these tests are inconclusive.

Alternating series

A series whose terms alternate sign can converge even when it is not absolutely convergent. If positive magnitudes decrease monotonically to zero, the alternating-series test guarantees convergence.

No single test works for every series. Choosing a test means recognizing the structure and long-term behavior of its terms.