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.

Note

When $s=2$, the series converges to $\frac{\pi}{6}$.

Geometric series

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

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

Alternating harmonic series

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

The above series is conditionally converging.

Special ones

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

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