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What Is Convergence And Ratio Test?

Referencing Styles : Harvard | Pages : 1
  • University: Trinity Western University

What is convergence?

Convergence is depicted as the technology that put focus or refers to the different systems that tends to move towards the performance of the similar tasks. Moreover, this concept deals with the involvement of the tasks that were previously performed by the different systems that have now converged within a single system which intends to perform all the tasks. The files in respect to the voice, video as well as data were previously transferred by the use of the different systems. However, this are now transmitted by the same communication protocols. The Internet has mostly helped or enhanced the communication methodology with the implementation of the convergence theory. Computer networks have allowed different operating systems that tends to communicate with the other computers present in the same network with the help of the same medium. Moreover, as the technological fields are being advanced in most of the sectors the trend is followed which eases the uses of convergence within this technologies. The simple concept in accordance to the theory of convergence involves the fact that it allows the multiple tasks that is to be performed in a single device that may tend to effectively conserve both the factor that is space as well as power. This is also a condition that depicts the state of the routers that may possess the same topological information about the internet work that they tend to operate. This theory is an important notion for the set of the routers that may engage in respect to the dynamic routing. All Interior Gateway Protocols tends to rely on the theory of convergence for proper functioning. The Exterior Gateway Routing Protocol BGP typically never converges because the Internet is too big for changes to be communicated fast enough.

What is ratio test?

The ratio test is depicted as a test that is implemented for the convergence of a series. It is shown by the following equation:

Where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The ratio test states that: if L < 1 then the series converges absolutely; if L > 1 then the series is divergent; if L = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.

The ratio test instructs us to find a limit we call ''L'':

There are three instructions implied:

  • form a ratio of an+1 divided by an 
  • take the absolute value of this ratio
  • take the limit as n → ∞

Once we find L, we then conclude:

  • if L < 1, the series converges
  • if L > 1, the series does not converge; it diverges 
  • if L = 1, the test is inconclusive and we can't tell from this test if the series converges or not

We can break down ratio testing into steps.

Step 1: Identify the general term, an

In our example,

Step 2: Find an expression for an+1

Simply replace the n in an with n+1:


Step 3: Build the ratio of an+1 to an.

Step 4: Simplify the ratio.

First, note division of fractions is pretty cumbersome. We can write division by a fraction as multiplication with the reciprocal of the fraction.

Then, organize the terms.

See how the ''2-to-the-power terms'' are written one over the other. Same for the n terms.

Further simplifying:

2n and 2n+1 have the same base of 2. This allows us to combine the exponents and express the division as a subtraction of powers: n + 1 - n.

But, n + 1 - n is just 1 and 21 is 1. Thus, the ratio becomes:


Step 5: Take the absolute value.

1/2 is clearly positive and n is always greater than 0 in the sum, so the absolute value of our ratio is just the ratio itself.

Step 6: Take the limit as n→∞.

The limit is on n. Since 1/2 does not depend on n, it comes out in front of the limit calculation.

How ratio test effects convergence. How is convergence used with series?

The topic of ''series convergence'' has specialized terminology and definitions. Below is provided an equation for the ratio test of the series convergence.


Where k is defined as the lower limit of a sum while an is known as the general term. The upper limit of the sum is depicted as n which tends to infinity.

The below example may state the overall explanation

In the above example, we can identify the k and the an. The k is 1 and

The capital sigma, Σ, is indicated as the summation notation that means to sum the general term from n = 1 to n = ∞. For further clarification  

Summing from n = 1 to n = 2:

But 21 is 2 and 22 is 4. Thus,


After, summing the first two terms we gain a result of 1.

After this we try summing up to n = 5:


From a sum equal to 1 using the first two terms, we now have a sum equal to 57/32 ≅ 1.78.

We could let M be the upper value for n. When we summed up to M = 2, we got a result of 1. When we summed up to M = 5, the result was 57/32. The idea of convergence is intuitive: if the series converges, the sum settles in to a finite value as M gets larger and larger. By the way, not all series will converge.

We could experiment by summing the series for increasing values of M. This would be a ''brute force'' approach. Intuitive but time-consuming and not very elegant. Instead, we can use a convergence test for determining convergence. There are many such tests. In this lesson, we explore convergence using the ratio test.

Derive both convergence and ratio test with formula.

When a sequence has a limit that exists, then the sequence is known as the convergent sequence. However, every sequence does not have any limits. For example, let us consider the sample sequence of the counting numbers:

  • 1, 2, 3, 4, . . .

If this sequence is stated to be continued, the terms increases in a certain specific manner, so an approaches infinity as n approaches infinity simultaneously. Therefore, the terms do not approach a number, as the value of infinity is not a number. Thus, the sequence doesn't have a limit as well as it is not convergent.

Now, for the determination of a sequence as a convergent one the below two steps are to be followed.

  1. Find a formula for the nth term, or an, of the sequence.
  2. Find the limit of that formula as n approaches infinity. If the limit exists, the sequence is convergent. If not, the sequence is not convergent.

The significant part of the derivation is stated to be the tricky one. Some formulas for sequences are obvious, but some are not. Consider the value sequence once again. At first instance, one may not be able to recognize a formula, but after having a critical look of the overall equation it can be said that the terms are of different type. The below figure depicts the stated statement

Looking at the above sequence an individual may find certain ways to write the an in terms of n. The first term is equal to 10,000/1, the second term is equal to 10,000/2, and so on. By continuing this methodology, we have that the nth term is equal to 10,000/n. We possess a formula for the nth term of the sequence:

an = 10,000/n 

The ratio test formula derivation


The proof will consist of the two parts

  1. Proof of 1 (if L < 1, then the series converges)
  2. Proof of 2 (if L > 1, then the series diverges)

Proof of 1 (if L < 1, then the series converges)

The primary aim is to compare the given series


With a convergent geometric series.

In this first case, L is less than 1, so we may choose any number r such that L < r < 1. Since


The ratio |an+1/an| will eventually be less than r. In other words, there exists an integer N such that



And, in general,



Is convergent because it is a geometric series with a common ratio r, such that 0 < r < 1. By the Comparison Test, the series



Therefore, our series is absolutely convergent.

Proof of 2 (if L > 1, then the series diverges)

The primary aim is to compare the given series


Therefore, the given series diverges by the Divergence Test.


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