Important Infinite Series

We can not define all the infinite series since there are infinite types of them. But the series below are very important so we need to remember them.

Geometric Series

A series of the form:

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}a{q}^{n-1}=a+aq+a{q}^2+a{q}^3\dots }$

When ${\displaystyle a\ne 0}$, we call it geometric series.

• When ${\displaystyle |q|<1}$, the geometric series converges.
• When ${\displaystyle |q|\ge 1}$, the geometric series diverges.

The sum of a geometric series ${\displaystyle S_n}$ is ${\displaystyle \frac{a(1-q^n)}{1-q}}$.

Harmonic Series

A series of the form:

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}}$

We call it harmonic series.

The harmonic series always diverge.

Collapsing Series

A series of the form like:

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n(n+1)}}$

We call it collapsing series.

The collapsing series usually converge.

${\displaystyle p}$ Series

A series of the form:

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^p}}$

We call it ${\displaystyle p}$ series.

• When ${\displaystyle p>1}$, the series converges.
• When ${\displaystyle p\le 1}$, the series diverges.

Actually, this is the general form of the harmonic series, when ${\displaystyle p=1}$, it is the harmonic series.

The ${\displaystyle p}$ series is very important in further study and applications.