Known Series

These series are helpful when using the direct comparison test or limit comparison test.

p-series

Not to be confused with power series.

Converges iff . When the series is called the harmonic series.

This series occurs in the definition of Riemann zeta function.

Note

When , the series converges to .

Geometric series

Converges iff . In that case, it converges to .

Alternating harmonic series

The above series is conditionally converging.

Special ones

Convergence of the above series can be proven using Dirichlet’s test.