# Mathematics - Mathematical Analysis Series Basics

Hello it's a me again drifter1! Today we continue with yet another post about **Mathematical Analysis** getting into **Series**. I will talk about some **Basics** to get started and we will get into more specific things posts to follow. So, let's get started!

## Getting into Series:

You all are familiar with the number π or pi that equals 3,1415.... But how did we end up with that number? How easy is it to get this number out of some Mathematical Equation?

Archimedes a mathematician of ancient Greece did some calculations and said that pi is in between of two rational numbers. 30/71 < pi < 3/7. But this doesn't give us a clear answer, but just speculates where pi is inside of a large numeral range.

A lot of mathematicians followed giving their own definition of pi as some form of Arithmetic Series.

For example Euler defined pi as:

So, defining pi is not easy and we can just approximately find a value of it using terms of a Series.

And this is mostly what Series are all about. They are about calculating the value/limit that a Series converges to when going to infinity! Each term of this Series/Sum can be anything and so a Sequence or many Sequences combined.

## Series Definition:

If (an) is a **sequence of real numbers** then we create a new sequence of real numbers like that:

where S(n) is called the sum of n terms of the sequence (an).

The **limit**/value that **this sum converges** to when n is going to infinity (**n -> ∞**) is called a **Series of real numbers**. So, a **Series **is **defined **as:

where the real numbers **a(n) for every natural number n are called the terms of the Series**.

The term **a(n) is called the generic term** of the Series, the same way as in sequences, but here each term is the value of a sequence.

Sometimes a **Series may not start from 1**, but from i != 1 and then we write the Series as:

A Series with positive terms is called a **positive Series** and a Series with negative terms a **negative Series**. A Series with alternating sign is called a **alternating Series**.

A Series S is bounded when the sum of n-terms (a1 + a2 + .... + an) is also bounded.

### Examples:

- The sum of all the natural numbers is a Series S(n) = 1 + 2 + 3 ... + n = n(n+1)/2 and is called a arithmetic Series
- The sum of all the natural numbers squared is a Series S(n) = 1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2n+)/6 and is commonly used in the mathematical induction method

### Series Convergence:

- A Series S
**converges**when the limit to infinity is equal to a real number s. - When the limit is equal to +-∞ then the sequence
**diverges**. - When the limit is undefined then the sequence "
**bounces**" - A Series S
**absolutely converges**when the series of absolutes values if the sequence (an) converges to a real number s

### Series Examples/Types:

**Geometric Series:**

A Series of the form:

with a being the first term and r the ratio.

- When |r| < 1 the series converges to s = a / (1-r)
- When r >= 1 the sequences diverges to +-∞
- When r <= -1 the sequence "bounces"

**Harmonic (Dirichlet) Series:**

A series of the form:

is called a p-harmonic series.

We will get into the convergence next time.

**Tylescoping Series:**

A series of the form:

where the generic term a(n) is written as a sub of two other terms of another sequence (bn).

To find out if it converges or not we check the convergence of (bn).

And this is actually all I wanted to cover today!

Next time we will get into how we find out if a Series converges or not using the so called Convergence Testing Criteria.

Bye!