Summation of Series
Introduction:
In sequences and series, the summation of series refers to the process of finding the sum of terms in a sequence up to a certain number of terms or to infinity. Various techniques and formulas exist to calculate the sum of specific types of series.
Notation:
The sigma ($\sum $) notation is commonly used to represent the summation of series:
$$\sum _{k=1}^{n}{a}_{k}={a}_{1}+{a}_{2}+\dots +{a}_{n}$$
Where:
 ${a}_{k}$ represents the terms of the series for $k$ ranging from $1$ to $n$.
 $n$ is the upper limit of summation.
Types of Series and Formulas:

Arithmetic Series:
 Sum of an arithmetic sequence.
 Formula: ${S}_{n}=\frac{n}{2}\times [2a+(n1)\times d]$, where ${S}_{n}$ is the sum of the first $n$ terms, $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms.

Geometric Series:
 Sum of a geometric sequence.
 Formula: ${S}_{n}=\frac{a\times (1{r}^{n})}{1r}$, where ${S}_{n}$ is the sum of the first $n$ terms, $a$is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Harmonic Series:
 Sum of a harmonic sequence.
 Harmonic series diverges to infinity: ${\sum}_{n=1}^{\mathrm{\infty}}\frac{1}{n}$.
Properties and Techniques:
Summation Properties:

Additive Property:
 ${\sum}_{k=1}^{n}[{a}_{k}+{b}_{k}]={\sum}_{k=1}^{n}{a}_{k}+{\sum}_{k=1}^{n}{b}_{k}$

Multiplicative Property:
 ${\sum}_{k=1}^{n}c\cdot {a}_{k}=c\cdot {\sum}_{k=1}^{n}{a}_{k}$

Changing the Index:
 ${\sum}_{k=m}^{n}{a}_{k}={\sum}_{j=m}^{n+p}{a}_{jp}$ where $p$ is an integer.

Partial Sums:
 Calculating the sum of a specific number of terms in a series.
 Useful for finding finite sums.

Infinite Series:
 Series that extend to an infinite number of terms.
 Determining convergence or divergence is essential.

Summation Rules:
 Rearranging terms, splitting series, using telescoping series, etc., to simplify calculations.

Convergence Tests:
 Various tests like the divergence test, ratio test, integral test, etc., to determine if an infinite series converges or diverges.
Applications:
 Mathematics: Calculating areas, volumes, and understanding limits.
 Physics: Modelling continuous phenomena, calculating total energy, etc.
 Engineering: Signal processing, control systems, etc.
 Finance: Compound interest, annuities, etc.