SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

                             CONVERGENCE TESTS, COMPARISON TEST, RATIO TEST, INTEGRAL TEST, POLYNOMIAL TEST, RAABE’S TEST

 


Given a particular series the first question one wishes to answer is whether the series converges or not. There is no single universal test that one can use to determine whether a series converges. Instead, there are a number of tests, some of which may be of use in one case, others in another.


Fundamental Principle. If a sequence Sn = φ(n) always increases as n increases but always remains less than a fixed number Q, then ole.gif exists and is not greater than Q.




Convergence tests



1. Comparison Test.


Convergence. A positive series is convergent if each of its terms is less than or equal to the corresponding terms of a series that is known to be convergent.


Divergence. A positive series is divergent if each of its terms is greater than or equal to the corresponding terms of a series that is known to be divergent.

 




2. Ratio Test. Let


             ole1.gif


Then the series Σun (positive or mixed-term)

 

            ● converges (absolutely) if L < 1 


            ● diverges if L > 1


If L = 1 the test fails.


If the test fails try the test of the following theorem.


Theorem 1. If, for a given series Σun,


             ole2.gif


ole3.gif


the series converges if b - a > 1 and diverges if b - a ole4.gif 1.


Example. Test the series


             ole5.gif


for convergence.


Solution. Using the ratio test


             ole6.gif


Thus the test is inconclusive.


Using Theorem 1,


             ole7.gif


and b - a = 3/2 - 1/2 = 1 


Therefore, by the theorem, the series diverges.



3. Integral Test. Let the general term of the series Σun be f(n), and let f(x) be the function obtained by replacing n by the continuous variable x. Now if for all values of x > a (where a is a positive integer) this function f(x) is positive and decreasing and if f(x) → 0 as x → ∞, then the series Σun converges if the integral


              ole8.gif


is convergent and diverges if this integral is divergent.



Example. Consider the series


             ole9.gif


Here

 

             ole10.gif


and


             ole11.gif


For all positive values of x this function is positive and decreasing, and as x → ∞, f(x) → 0 . In addition


             ole12.gif ole13.gif


Thus the integral converges and the series is convergent.

 

                                                                                                

 


4. Polynomial test. If un = g(n)/h(n) where g(n) and h(n) are polynomials in n, then the series Σun is convergent if the degree of h(n) exceeds that of g(n) by more than 1; otherwise the series is divergent.


Example. Consider the series


             ole14.gif


Here


             ole15.gif


The degree of the denominator exceeds that of the numerator by 2 and therefore the series is convergent.

 



5. Alternating series test. An alternating series Σun is convergent if


             ole16.gif


and


             ole17.gif


 



6. The root test. Let


             ole18.gif


Then the series Σun (positive or mixed-term)

 

            ● converges (absolutely) if L < 1 


            ● diverges if L > 1


If L = 1 the test fails.


 



7. Raabe’s test. Let


             ole19.gif


Then the series Σun (positive or mixed-term)

 

            ● converges (absolutely) if L > 1 


            ● diverges or converges conditionally if L < 1


If L = 1 the test fails.


 

Raabe’s test is often used when the ratio test fails.





References

  Middlemiss. Differential and Integral Calculus.

  Ayres. Calculus (Schaum).

  Oakley. The Calculus (COS).

  Spiegel. Advanced Calculus (Schaum).




More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]