# 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:

1. Arithmetic Series:

• Sum of an arithmetic sequence.
• Formula: ${S}_{n}=\frac{n}{2}×\left[2a+\left(n-1\right)×d\right]$, 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.
2. Geometric Series:

• Sum of a geometric sequence.
• Formula: ${S}_{n}=\frac{a×\left(1-{r}^{n}\right)}{1-r}$, 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.
3. Harmonic Series:

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

#### Summation Properties:

• ${\sum }_{k=1}^{n}\left[{a}_{k}+{b}_{k}\right]={\sum }_{k=1}^{n}{a}_{k}+{\sum }_{k=1}^{n}{b}_{k}$
2. Multiplicative Property:

• ${\sum }_{k=1}^{n}c\cdot {a}_{k}=c\cdot {\sum }_{k=1}^{n}{a}_{k}$
3. Changing the Index:

• ${\sum }_{k=m}^{n}{a}_{k}={\sum }_{j=m}^{n+p}{a}_{j-p}$ where $p$ is an integer.
1. Partial Sums:

• Calculating the sum of a specific number of terms in a series.
• Useful for finding finite sums.
2. Infinite Series:

• Series that extend to an infinite number of terms.
• Determining convergence or divergence is essential.
3. Summation Rules:

• Rearranging terms, splitting series, using telescoping series, etc., to simplify calculations.
4. 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.