# D'Alembert's Ratio Test of Convergence of Series

In this article, we will formulate the D' Alembert's Ratio Test on convergence of a series.

Let's start.

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## Statement of D'Alembert Ratio Test

A series $ \sum {u_n}$ of positive terms is convergent if from and after some fixed term $ \dfrac {u_{n+1}} {u_n} < r < {1} $ , where r is a fixed number. The series is divergent if $ \dfrac{u_{n+1}} {u_n} > 1$ from and after some fixed term.

D'Alembert's Test is also known as the **ratio test of convergence of a series**.

**Theorem**

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers in $R$, or a series of complex numbers in $C$.

Let the sequence ${a_n}$ satisfy:

$\displaystyle \lim_{n \mathop \to \infty} {\frac {a_{n + 1} } {a_n} } = l$

- If $l > 1 $, then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges.
- If $l < 1 $, then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.

## Definitions for Generally Interested Readers

**(Definition 1)** An infinite series $ \sum {u_n}$ i.e. $ u_1 + u_2 + u_3 + \ldots + u_n$ is said to be **convergent** if $ S_n$ , the sum of its first $ n$ terms, tends to a finite limit $S$ as n tends to infinity.

We call $S$ the sum of the series, and write $$ S=\displaystyle {\lim_{n \to \infty} } S_n$$ .

Thus an infinite series $ \sum {u_n}$ converges to a sum S, if for any given positive number $ \epsilon $ , however small, there exists a positive integer $ n_0$ such that $ |S_n-S| < \epsilon$ for all $ n \ge n_0$ .

**(Definition 2)**

If $ S_n \to \pm \infty$ as $ n \to \infty$ , the series is said to be **divergent**.

Thus, $ \sum {u_n}$ is said to be divergent if for every given positive number $ \lambda$ , however large, there exists a positive integer $ n_0$ such that $ |S_n|>\lambda$ for all $ n \ge n_0$ .

**(Definition 3)**

If $ S_n$ does not tend to a finite limit, or to plus or minus infinity, the series is called **oscillatory**.

## Proof & Discussions on Ratio Test

Let a series be $ \mathbf {u_1+u_2+u_3+.......}$ . We assume that the above inequalities are true.

From the first part of the statement:

$ \dfrac {u_2}{u_1} < r$ , $ \dfrac {u_3}{u_2} < r $ ......... where r <1.

Therefore $$ {u_1+u_2+u_3+....}$$

$$= u_1 {(1+\frac{u_2}{u_1}+\frac{u_3}{u_1}+....)}$$

$$ =u_1{(1+\frac{u_2}{u_1}+\frac{u_3}{u_2} \times \frac{u_2}{u_1}+....)} $$

$$ < u_1(1+r+r^2+.....)$$

Therefore, $ \sum{u_n} < u_1 (1+r+r^2+.....)$

or, $ \sum{u_n} < \displaystyle{\lim_{n \to \infty}} \dfrac {u_1 (1-r^n)} {1-r}$

Since r<1, therefore as $ n \to \infty , \ r^n \to 0$

therefore $ \sum{u_n} < \dfrac{u_1} {1-r}$ =k say, where k is a fixed number.

Therefore $ \sum{u_n}$ is convergent.

Since, $ \dfrac{u_{n+1}}{u_n} > 1$ then, $ \dfrac{u_2}{u_1} > 1$ , $ \dfrac{u_3}{u_2} > 1$

Therefore

$ u_2 > u_1$

$u_3 >u_2>u_1$

$u_4 >u_3 > u_2 >u_1$

and so on.

Therefore $ \sum {u_n}=u_1+u_2+u_3+....+u_n$ > $ nu_1$ .

By taking n sufficiently large, we see that $ nu_1$ can be made greater than any fixed quantity.

Hence the series is divergent.

**Academic Proof**s

From the statement of the theorem, it is necessary that $\forall n: a_n \ne 0$; otherwise ${\dfrac {a_{n + 1} } {a_n} }$ is not defined.

Here, ${\dfrac {a_{n + 1} } {a_n} }$ denotes either the absolute value of $\dfrac {a_{n + 1} } {a_n}$, or the complex modulus of $\dfrac {a_{n + 1} } {a_n}$.

#### Absolute Convergence

Suppose $l < 1$.

Let us take $\epsilon > 0$ such that $l + \epsilon < 1$.

Then:

$\exists N: \forall n > N: {\dfrac {a_n} {a_{n - 1} } } < l + \epsilon$

Thus: $ (\displaystyle {a_n}) $

$= (\displaystyle {\frac {a_n} {a_{n - 1} } } {\frac {a_{n - 1} } {a_{n - 2} } } \dotsm {\frac {a_{N + 2} } {a_{N + 1} } } {a_{N + 1} })$

$< (\displaystyle {l + \epsilon}^{n - N - 1} {a_{N + 1} })$

By **Sum of Infinite Geometric Progression**, $\displaystyle \sum_{n \mathop = 1}^\infty{l + \epsilon}^n$ converges.

So by the corollary to the comparison test, it follows that $\displaystyle \sum_{n \mathop = 1}^\infty {a_n}$ converges absolutely too.

$\blacksquare$

#### Divergence

Suppose $l > 1$.

Let us take $\epsilon > 0$ small enough that $l - \epsilon > 1$.

Then, for a sufficiently large $N$, we have:

$(\displaystyle {a_n}) =$

$(\displaystyle {\frac {a_n} {a_{n - 1} } } {\frac {a_{n - 1} }{a_{n - 2} } } \dotsm {\frac {a_{N + 2} } {a_{N + 1} } } {a_{N + 1} })$

$> (\displaystyle {l - \epsilon}^{n - N + 1} {a_{N + 1} })$

But ${l - \epsilon}^{n - N + 1} {a_{N + 1} } \to \infty$ as $n \to \infty$.

So $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges.

$\blacksquare$

## Comments

- When $ \dfrac {u_{n+1}} {u_n}=1$ , the test fails.
- Another form of the test-- The series $ \sum {u_n}$ of positive terms is convergent if $ \displaystyle {\lim_{n \to \infty}} \dfrac {u_n}{u_{n+1}}$ >1 and divergent if $ \displaystyle{\lim_{n \to \infty}} \dfrac {u_n}{u_{n+1}}$ <1.
- One should use this form of the test in the practical applications.

## Suggested Reading

### Analysis I, Fourth Edition

Author: Terence Tao

**Hardcover : **376 pages

## An Example

Verify whether the infinite series $ \dfrac{x}{1.2} + \dfrac {x^2} {2.3} + \dfrac {x^3} {3.4} +....$ is convergent or divergent.

## Solution

We have $ u_{n+1}= \dfrac {x^{n+1}}{(n+1)(n+2)}$ and $ u_n= \dfrac {x^n} {n(n+1)}$

Therefore $ \displaystyle {\lim_{n \to \infty}} \dfrac{u_n} {u_{n+1}} = \displaystyle{\lim_{n \to \infty}} (1+\frac{2}{n}) \frac{1}{x} = \frac{1}{x}$

Hence, when 1/x >1 , i.e., x <1, the series is convergent and when x >1 the series is divergent.

When x=1, $ u_n=\dfrac{1} {n(n+1)}=\dfrac {1}{n^2} {(1+1/n)}^{-1}$

or, $ u_n=\dfrac{1}{n^2}(1-\frac{1}{n}+ \frac {1}{n^2}-.....)$

Take $ \dfrac{1}{n^2}=v_n$ Now $ \displaystyle {\lim_{n \to \infty}} \dfrac {u_n}{v_n}=1$ , a non-zero finite quantity.

But $ \sum {v_n}=\sum {\frac{1}{n^2}}$ is convergent.

Hence, $ \sum {u_n}$ is also convergent.