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?

This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

Introduction to Series

Arithmetic Series

Example

Geometric Series

Example:

Real-World Applications