country
$20 Bonus + 25% OFF
Securing Higher Grades Costing Your Pocket? Book Your Assignment at The Lowest Price Now!
Add File

Error goes here

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:

Simplifying:


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

Proof

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

If


Therefore, the given series diverges by the Divergence Test.

Solutions

OR

MyAssignmenthelp.com is diligent to fulfil students needs in the best possible manner by providing both thesis writing services and thesis editing services. We boast separate teams of thesis writing experts and editors who handle the students requests like can anyone do editing of my thesis online or can someone write my thesis online. Thus, we have gained the status of being the most efficient and reliable thesis help provider company in USA.

Save Time & improve Grade

Just share requirement and get customized Solution.

watch
question
We will use e-mail only for:

arrow Communication regarding your orders

arrow To send you invoices, and other billing info

arrow To provide you with information of offers and other benefits

Add File

Error goes here

1,423,472

Orders

4.9/5

Overall Rating

5,092

Experts

Our Amazing Features

delivery

On Time Delivery

Our writers make sure that all orders are submitted, prior to the deadline.

work

Plagiarism Free Work

Using reliable plagiarism detection software, Turnitin.com.We only provide customized 100 percent original papers.

time

24 X 7 Live Help

Feel free to contact our assignment writing services any time via phone, email or live chat. If you are unable to calculate word count online, ask our customer executives.

subject

Services For All Subjects

Our writers can provide you professional writing assistance on any subject at any level.

price

Best Price Guarantee

Our best price guarantee ensures that the features we offer cannot be matched by any of the competitors.

Our Experts

Assignment writing guide
student rating student rating student rating student rating student rating 5/5

1265 Order Completed

97% Response Time

James Cook

Masters in Management

Wellington, New Zealand

Hire Me
Assignment writing guide
student rating student rating student rating student rating student rating 5/5

647 Order Completed

98% Response Time

Adlina Han

Masters in Marketing with Specialization in Branding

Singapore, Singapore

Hire Me
Assignment writing guide
student rating student rating student rating student rating student rating 5/5

2830 Order Completed

97% Response Time

Leroy Bicknell

MBA in Marketing

London, United Kingdom

Hire Me
Assignment writing guide
student rating student rating student rating student rating student rating 5/5

203 Order Completed

97% Response Time

Richard Alpert

PhD in Psychology

London, United Kingdom

Hire Me

FREE Tools

plagiarism

Plagiarism Checker

Get all your documents checked for plagiarism or duplicacy with us.

essay

Essay Typer

Get different kinds of essays typed in minutes with clicks.

edit

GPA Calculator

Calculate your semester grades and cumulative GPa with our GPA Calculator.

referencing

Chemical Equation Balancer

Balance any chemical equation in minutes just by entering the formula.

calculator

Word Counter & Page Calculator

Calculate the number of words and number of pages of all your academic documents.

Refer Just 5 Friends to Earn More than $2000

Check your estimated earning as per your ability

1

1

1

Your Approx Earning

Live Review

Our Mission Client Satisfaction

Great service! Got a really good grade for it.Even my teacher was proud of it,great expressing and gramar.

flag

User Id: 413774 - 23 Sep 2020

Australia

student rating student rating student rating student rating student rating

Professional writing with zero errors. I got a 92% grade on my project management assignment.

flag

User Id: 345729 - 23 Sep 2020

Australia

student rating student rating student rating student rating student rating

This assignment was returned several times for corrections and the appropriate correction was not provided by writer. I use Grammarly and found a lot of mistakes. I have to rewrite the whole assignment. very disappointed.

flag

User Id: 359164 - 23 Sep 2020

Australia

student rating student rating student rating student rating student rating

Help assignment is really helpful site who are comfortable to use it , expert are very well educated and work is done in timely manner

flag

User Id: 409673 - 23 Sep 2020

Australia

student rating student rating student rating student rating student rating
What Is Convergence And Ratio ... has been added in your library.
callback request mobile
Have any Query?