Summary  
This chapter explains how to implement code to calculate the sum of arithmetic and geometric series using closed-form formulas (including handling infinite geometric series convergence).

General domain of usage  
Financial analysis

**A series** is a mathematical expression formed by adding the terms of a sequence. The most common types are the **arithmetic series** and the **geometric series**, which differ in how their terms progress.


Definition

## Arithmetic Series

An **arithmetic series** is formed when the difference between consecutive terms in a sequence is constant.

$$
2, 5, 8, 11, 14, ...; (\text{common difference}, d = 3)
$$

The sum of the first $$n$$ terms of an arithmetic series is given by:

$$
S_n = \frac{n}{2} \cdot (a + l)
$$

Where:
- $$n$$ - number of terms;
- $$a$$ - first term;
- $$l$$ - last term.

Alternatively, if the last term $$l$$ is not known:

$$
S_n = \frac{n}{2} \cdot \left( 2a + (n - 1) \cdot d \right)
$$

### Example

Find the sum of the **first 10** terms of the series $$2,5,8,...$$

$$
S_{10} = \frac{10}{2} \cdot (2 + (10 - 1) \cdot 3) = 5 \cdot (2 + 27) = 145
$$

## Geometric Series

A **geometric series** is formed when each term in the sequence is multiplied by a fixed ratio to get the next term.
 
$$
3,6,12,24,48,...;(\text{common ratio}, r=2)
$$ 


The sum of the first $$n$$ terms of a geometric series is given by:
 
$$
S_n = a \cdot \frac{1 - r^n}{1 - r},\ r \neq 1
$$

Where:
- $$a$$ - first term;
- $$r$$ - common ratio;
- $$n$$ - number of terms.

If the series is **infinite** and $$|r|<1$$:

$$
S = \frac{a}{1 - r}
$$


### Example:
Find the sum of the **first 4** terms of the series $$3,6,12,24,...$$

$$
S_4 = 3 \cdot \frac{1-2^4}{1-2} = 3 \cdot \frac{1-16}{-1}=3 \cdot 15 = 45
$$


## Real-World Applications

Arithmetic and geometric series appear in many data science contexts:

* **Population growth** and **resource modeling** through geometric progressions;
* **Financial analysis** using compound interest calculations;
* **Revenue forecasting** across time periods;
* **Machine learning**, where summations occur in algorithms like gradient descent.


$$a=1$$, $$r=0.5$$ and $$n=\infty$$, what is the sum of the infinite geometric series?

Master the mathematical foundations essential for data science. Explore core concepts in functions, calculus, linear algebra, probability, and dimensionality reduction. Build both theoretical understanding and practical coding experience to strengthen your ability to analyze data, model complex systems, and apply advanced techniques in machine learning.

Explore the foundation of mathematical functions. Learn different types of algebraic and transcendental functions, their properties, and how to implement them in Python to solve real-world problems.

Master the concepts of sets and series, from basic operations to practical applications. Gain hands-on experience implementing set operations and working with arithmetic and geometric series in Python.

Develop a solid understanding of limits, derivatives, integrals, and partial derivatives. Connect theory to practice by implementing these concepts in Python and applying them to optimization through gradient descent.

Build strong knowledge of vectors, matrices, and transformations. Learn decomposition methods and eigenvalue analysis, while reinforcing concepts with Python coding challenges and practical data science applications.

Dive into probability theory and statistics. Study conditional probability, Bayes' theorem, and statistical measures. Implement key concepts in Python, simulate distributions, and strengthen your skills through challenges and quizzes.

Introduction to Series

Arithmetic Series

Example

Geometric Series

Example:

Real-World Applications