Series

Given a sequence it's sum sequence can be written like this:

The pair made out of a sequence and its sum is called a series of general term . It can be expressed as .
Examples:

is convergent/divergent/oscillating if the partial sum sequence () is. Note that is also used to denote the

#todo add sum properties

Convergence criteria

Given a sequence we can denote its sum sequence as

Cases based of

values

A p-series is a series defined as

Cases based of

values

Hidden geometric series

A series like might seem like something related to p-series at first but is actually a geometric series with :

General idea

The general idea of the comparison criteria is given a series of the sequence () find a different sequence which is either:

Convergent and is above after a certain point
Divergent and is below after a certain point
After that it's fairly easy to understand why would be either convergent (1) or divergent (2)
Since is above and convergent cannot surpass it and thus must be also convergent.
Since is below and divergent cannot drop beneath it and must also be divergent.

Consider 2 series and

Consider 2 series and
We can calculate

If then both series have the same character (either both diverge or both converge)
If and converges, then diverges
If and diverges, then converges
Info

The point here is to select which you know converges or diverges and which, when calculating results in one of the cases.

Consider the series
We can study

Consider the series
If both of the listed conditions are true:

The sign of the term alternates
The sequence is always decreasing and converges at 0
Then the series is convergent

For series of type if and are polynomials with simple roots we can separate the polynomials into fractions:

Example

Therefore

After decomposing the initial sequence into several smaller fractions we can rewrite the series as a sum of several series.

Example

This series sum can be written out with offsets in a way such that the divisors in each column are equal

Example