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.

\sum_{k=1}^\infty \frac{1}{k^s}

Converges iff $s \gt 1$ . When $s=1$ the series is called the harmonic series.

This series occurs in the definition of Riemann zeta function.

\sum_{k=1}^\infty r^k

Converges iff $\lvert r \rvert \lt 1$ . In that case, it converges to $\frac{1}{1-r}$ .

\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} = \ln 2

The above series is conditionally converging.

\sum_{k=1}^\infty \frac{\sin k}{k}\;\text{is converging}

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

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September 13, 2026 1 min read

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