Infinite Series and Sequences

[For Prose]

I am having difficulty with infinite series and sequences within calculus. I am not quite sure where the difficulty is stemming from. I understand the general idea but when it comes to knowing when to utilize the comparison test, divergence test, integral test, ect., I cannot seem to make things click.

 

I am hoping for a user friendly explanation.

 

If this is too vague, I am happy to clarify specifics.

[Unity+]

I am having difficulty with infinite series and sequences within calculus. I am not quite sure where the difficulty is stemming from. I understand the general idea but when it comes to knowing when to utilize the comparison test, divergence test, integral test, ect., I cannot seem to make things click.

 

I am hoping for a user friendly explanation.

 

If this is too vague, I am happy to clarify specifics.

I am confused what you are asking for.

 

Basically, the comparison test is used to determine, especially fractions, whether the fraction will result in a number, 0, or infinity. The divergence test is a test to determine if the problem will diverge, where it goes off to infinity, or if it will converge, or eventually become a specific value.

 

And for Integral test:

 

 

Suppose that f(x) is a continuous, positive and decreasing function on the interval and that f(n) = a_n then,

1. If is convergent so is .
2. If is divergent so is .

http://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx

[For Prose]

What I am looking for is an explanation using only words. Specifically, why limits and integrals play a role and how you can determine just by looking at a series or sequence what steps you will need to take to determine whether or not it converges.

[Unity+]

What I am looking for is an explanation using only words. Specifically, why limits and integrals play a role and how you can determine just by looking at a series or sequence what steps you will need to take to determine whether or not it converges.

Limits play a role in calculus because of what role they play in determining the derivative(the tangent of a curve) or other equations that originally would involve being undefined(i.e, having a denominator of 0).

 

How they work with derivatives is given by the following equation(sorry if an equation is being used).

 

The idea here is you have a secant line and the point is to result in having a tangent line. This involves the variable approaching 0. Here is a graph below to demonstrate what it essentially is.

Let me break down the equation for you. If you look at the top part, you will notice that it is f(x+h)+f(x). This is part of the definition of secant line, which is simply the change in y over the change in x.

 

Of course, this only gets the slope of the SECANT line, not the tangent line. Limits become important in finding the tangent line because the limit is used to find what the slope of the tangent line will be AT that particular point x instead of between the two points. As the change in x approach 0, the secant line becomes the tangent line.

 

If you were to analyze this without the use of calculus, you would realize that the regular equation, the change in y over the change in x, would not work because if there is no change in x, there is no change in y. This means it would be 0/0, which is not a good thing to get in Mathematics.

 

Integrals are significant not only because they are the result of the anti-derivative, but also because they allow for the calculation of a curve. They mostly play a role in Physics, but have uses in many other fields.

 

 

 

how you can determine just by looking at a series or sequence what steps you will need to take to determine whether or not it converges.

In relation to derivatives, limits, and integrals the only thing that there is to test is whether the limit exists or if the derivative exists at a particular point, for that matter. That involves determining if there is discontinuous parts of the graph and other factors. If you mean in with sequences or summations, it depends on the equation you are dealing with.

 

Here is a wiki page that can show some tricks: http://www.wikihow.com/Determine-Whether-an-Infinite-Series-Converges

[studiot]

You should study sequences before you study series; they are simpler and it doesn't take long, have you done this?

Your understanding of what convergence is, is also key.

 

Can you say if the following sequence is convergent or divergent?

 

1,1,1,1,1,1,1,1..........................................1,1,1,..................1,1,1,.......

Edited November 2, 2014 by studiot

[For Prose]

Here is a wiki page that can show some tricks: http://www.wikihow.com/Determine-Whether-an-Infinite-Series-Converges

 

This brought me slightly closer to understanding Unity.

 

You should study sequences before you study series; they are simpler and it doesn't take long, have you done this?

Your understanding of what convergence is, is also key.

 

Can you say if the following sequence is convergent or divergent?

 

1,1,1,1,1,1,1,1..........................................1,1,1,..................1,1,1,.......

 

Divergent.

 

I have studied series and sequences for quite some time now and am only having trouble putting the mathematical concepts into words. Usually, I am able to explain something to others but cannot do so when it comes to deciphering series and sequences. I know all of the definitions and have access to all the information regarding their notation. But I think it is the notation that is causing the confusion.

 

I can easily memorize the rules but if I don't understand the whys I will fail miserably in my upper level mathematics courses.

Also, I understand convergence to be to some finite number whereas, divergence, a limit DNE.

[Unity+]

Divergent.

Actually, it is convergent. Correct me if I am wrong studiot.

 

Because the series doesn't go off to infinity and simply results to 1, it converges on the value 1.

[studiot]

That is a very good point unity+. (+1)

 

This is where I was trying to lead. For Prose you may have seen some expositions of convergenge but have you seen this one?

 

First the idea is that the controlling definition is convergence.

 

If a sequence or series is not convergent it is divergent..

 

So the tests are basically (nearly) all for convergence.

 

Now a definition of convergence is meant to convey the idea that, after some the Nth term in any sequence every term is within a certain distance of the limit.

 

so for any term m > N : (T_L - T_m ) <= some delta, however small.

 

Or to put it another way a sequence is convergent if for any delta > 0 There is a T_N such that every Tm where m>N is less than delta.

 

Using this definition I would say that Unity+ is correct.

 

Equally, by definition any sequence that does not satisfy this is divergent. So for instance if I had alternated +1, -1, +1. -1 etc then the above condition would not be met.

 

Does this help ?

Edited November 3, 2014 by studiot

[For Prose]

My response was to a series studiot, not a sequence. My fault.

 

I found today what I was missing. They are helpful in solving equations for which integration cannot be performed. I also learned that a harmonic series refers only to one specific series, not a collection.

 

Also, the reason for determining partial sums was also made clearer. I found it strange that I was having so much difficulty, but found out from one of my math professors that entire classes are spent on dealing with these topics and that my initial confusion is somewhat warranted.

[studiot]

Yeah, series are very important.

 

Look in you university library for the book

 

Infinite Series

by

Fort

Oxford University Press

[For Prose]

I looked for it today and could not find it but will continue my search to the other libraries around town.

 

Thanks for the input!