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# Diverge|Definition & Meaning

## Definition

The **divergent sequence** is the one in which the words **never** come close to zero. they **persist** to raise or decline and they loom **infinity** as n comes **infinity.**

**Diverge **(in the context of sequences and series) means not **settling towards** some value. In other words, **not converging** to a fixed value. When a series (sum of all elements in a sequence) diverges, it goes to **positive** or **negative infinity** as the number of elements in the sequence approaches infinity**.**

The particular **phrases** of the **series** must **verge zero** if a series **converges.** Therefore any **series** in which the **respective representations** do not approach zero, subsequently **diverge.** Regardless, **convergence** is a more **powerful** condition. There is a possibility that not all series whose terms approach zero converge. A **harmonic series** is a very suitable example.

**Nicole Oresme,** a medieval mathematician, proved the **divergence** of the harmonic series

## Infinite Series

Any **ordered infinite sequence** as in current nomenclature, $(a_1 , a_2 , a_3 ,\space … )$ of **representations** (that is functions, numbers, or anything that can be counted) explains a series, which is the **operation** of **summing** the $a_i$ one after the other.** Because** of the fact that there are an infinite number of **terms **in the series, it is called an **Infinite Series**. Infinite series is described(or characterized) with an **expression** like:

The infinite sequence of additions that are **denoted** by a series cannot be **virtually** taken on (at the smallest in a finite portion of time). Nevertheless, it is **occasionally attainable** to transfer a value to a series if the set from which the numbers and finite **aggregates** of numbers belong has an **idea** of limit, named the sum of the series.

This **value** is the **boundary** as n overlooks to infinity (if the limitation persists) of the addition of the $n$ first instances of the series, that are named the nth partial sums of the series. That is described as:

\[ \sum_{i=1}^{\infty} = lim_{n \rightarrow \infty} \sum_{i=1}^{n} a_i \]

One states that the series is **convergent** or **summable** if this limit persists, or if the arrangement $(a_1, a_2, a_3, … )$ is summable. In this matter, the limit is named as the **sum** of the **series.** If that’s not the case, the series is said to be **divergent.**

The following notation $\sum_{i=1}^{\infty} a_i $ **characterizes** both the series, that is the implied method of summing the terms of the series **indefinitely. **if the series converges, the **outcome** of the **procedure** is the sum of the series. This is a conception of the **identical** ritual of indicating by $a + b$ both the addition, and the process of adding, and its **outcome** is the sum of a and b.

## Theorems on Methods for Summing Divergent Series

A **summability** procedure denoted as M is standard if it coordinates with the authentic limit on all convergent series. Such a derivative is anointed an **Abelian theorem** for M, derived from the prototypical Abel’s theorem. More subtle in nature, called **Tauberian theorems,** partial converse results from a prototype proved by **Alfred Tauber.**

Here, **partial** **converse** describes that if M sums the series $\sum$, and some side-condition maintains, then $\sum$ was **convergent** in the foremost position; without any side-condition, this type of **outcome** would state that M only added **convergent** series (making it worthless as an **addition** method for divergent series).

The operation providing the **aggregate** of a convergent series is linear, and it pursues from the **Hahn–Banach theorem** that it can be expanded to a summation process summing any series with fixed partial sums. This is known as the **Banach limit.**

This truth is not very helpful in training, as there are many such **attachments,** fluctuating with each other, requiring invoking the **axiom** of choice or its equivalents, and also proving such operators exist such as **Zorn’s lemma.** They are consequently **not constructive** in nature.

The topic of divergent series, as a discipline of mathematical computation, is mainly discussed with **detailed** and **realistic** procedures such as **Abel** summation, **Cesàro summation,** and **Borel summation,** and their relationships. The beginning of **Wiener’s tauberian theorem** sketched an approach to the subject, presenting unplanned associations to **Banach algebra techniques** in Fourier calculation.

## Convergent vs. Divergent Series

A **convergent** series also called a **limit,** is a series whose partial sums lean to a distinctive number. A **divergent** series by difference is a series whose partial sums, **don’t come** to a **limit.** Divergent series traditionally go to $\infty$, go to $- \infty$, or don’t come to one specific number.

A simple example of a **convergent** series is as follows:

The **Partial** sums glimpse like this:

\[ \dfrac{1}{2}, \dfrac{3}{4}, \dfrac{7}{8}, \dfrac{15}{16}, \dots \]

and we can also notice that they reach nearer and nearer to 1. The foremost partial sum is $\dfrac{1}{2}$ away, the second $\dfrac{1}{4}$ away, and so on and so forth until it is infinitely near to 1. This means that the above series **converges** to a limit of **1.**

The **convergence** of various series is a merely suitable issue in higher math and multiple important **mathematical** findings rotate around various series, but for the **intentions** of **Algebra 2,** we can specifically involve ourselves with that whether or not **geometric** series will **converge.**

Provided a **geometric** series with a **standard** ratio $r$, it will **converge** whenever $|r|<1$. If we had a geometric series with a common ratio of say, 2, this should create **instinctive sense,** we would have something that peeks like:

\[ \sum_{n=0}^{\infty} 2^n = 1+2+4 +8+16+ \dots \]

Indeed, the **numbers** we are adding to the sum get bigger and bigger, so the **sum** would never be restricted down and comes to a specific limit. Following is an example of a **divergent** series.

Again, if the common ratio is −2, the series:

\[ \sum_{n=0}^{\infty} (-2)^n = 1-2+4 -8+16-32+\dots. \]

will also diverge. A better way to see this is:

\[ \sum_{n=0}^{\infty} (-2)^n = (1-2)+ (4-8)+ (16-32) + \dots \]

\[ = -1 -4 -16 \]

\[ = – \sum_{n=0}^{\infty} 4^n \]

which is absolutely **divergent**.

## An Example of Determining if a Series Diverges

Find out if the series given below is divergent or convergent.

\[ \sum_{n=1}^{\infty} n\]

### Solution

To identify if the series is **convergent,** we are first instructed to use the formula:

\[ s_n = \sum_{i=1}^{n} i\]

The **value** of the series can be:

\[ s_n = \sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}\]

So, to **find out** if the series is converging, we are preferably demanded to examine the **arrangement** of partial sums:

\[ \left. \frac{n(n+1)}{2} \right\rvert_{n=1}^{\infty} \]

\[ lim_{n \rightarrow \infty } \dfrac{n(n+1)}{2} = \infty \]

**Accordingly,** the sequence of partial sums **diverges,** and consequently, the series also **diverges.**

*All images/mathematical drawings were created with GeoGebra.*