a) Harmonic series is the divergent infinite series:

\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{1}}{{n}}={1}+\frac{{1}}{{2}}+\frac{{1}}{{3}}+\frac{{1}}{{4}}+\frac{{1}}{{5}}+…\)

b) Alternating harmonic series

It’s converging series, which converge to \(\displaystyle{{\ln}_{{2}}}\)

\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{\left(-{1}\right)}^{{{n}–{1}}}}{{n}}={\ln{{2}}}\)

\(\displaystyle{1}+\frac{{1}}{{3}}+\frac{{1}}{{5}}+…={2.021}\)

c) Geometric series is a series with a constant ratio between successive terms:

\(\displaystyle\frac{{1}}{{2}}+\frac{{1}}{{4}}+\frac{{1}}{{8}}+\frac{{1}}{{16}}+…\)

\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{1}}{{n}}={1}+\frac{{1}}{{2}}+\frac{{1}}{{3}}+\frac{{1}}{{4}}+\frac{{1}}{{5}}+…\)

b) Alternating harmonic series

It’s converging series, which converge to \(\displaystyle{{\ln}_{{2}}}\)

\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{\left(-{1}\right)}^{{{n}–{1}}}}{{n}}={\ln{{2}}}\)

\(\displaystyle{1}+\frac{{1}}{{3}}+\frac{{1}}{{5}}+…={2.021}\)

c) Geometric series is a series with a constant ratio between successive terms:

\(\displaystyle\frac{{1}}{{2}}+\frac{{1}}{{4}}+\frac{{1}}{{8}}+\frac{{1}}{{16}}+…\)