Calculus II - Determining if Infinite Sequences Converge

In summary: I hadn't seen that, but it is a good source for practice problems and many of them are worked out. It's still not a "gigantic list" of problems with answers.In summary, the conversation was about studying infinite series and the request for a list of series and proofs of convergence or divergence. The expert advises using formula sheets for calculus and learning when to apply different tests rather than memorizing a list of series. They also suggest developing a personal algorithm for determining which test to use. The conversation concludes with the reminder that experience and practice are important in mastering this skill and that there is no one definitive source for a "gigantic list" of series problems with worked-out tests.
  • #1
GreenPrint
1,196
0
Hi,

I'm studying infinite series and was wondering if someone could recommend me a gigantic list of examples of series and proofs of weather they converge or not.
 
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  • #2
http://prairiestate.edu/skifowit/math/

Have a look at formula sheets for calculus.
 
  • #3
It's called your book!

There are infinitely many series that converge and diverge in your study of infinite series...just saying...
 
  • #4
I can't find a list on there of series and proofs of why they converge or not.
 
  • #5
My book only has so many examples and I did all the homework and don't feel like solving more problems and just woundering if someone could recommend a good list that I could look at of many series that I could just go over that is all
 
  • #6
What you are looking for are probably tests not proofs. Use the link I provided.
 
  • #7
http://prairiestate.edu/skifowit/sheets/c05.pdf

http://prairiestate.edu/skifowit/sheets/c10.pdf
 
  • #8
ya i saw those but i was like looking for a list of a bunch of series and proofs for why they converge or not because I'm tried of solving problems and think it would be more efficient to just look at a already made list of series and proofs for why they converge or not so that way i can be so familiar with infinite series that I don't have to think twice about which test to use
 
  • #9
Like I said you are looking for tests not proofs. Proofs would be much more challenging then simply applying theorems. I would say that knowing when to use various tests in much more efficient than trying to memorize a list of series. By the way, when I took Calculus II I developed a fairly simple algorithm to determine which test to use. See if you can come up with one. It is more convenient to know why the test work and when to apply it than to memorize series.
 
  • #10
The only problem is that
I don't think there's one algorithm to use that always works?
And I have a general idea of when to use which test, but am kind of bad at coming up with comparisons for the limit comparison test, your suppose to pick the comparison to be some function that represents the original function as n goes to infinity or like behaves like it?
and you I have memorized the tests but was just like looking for a giant list of examples with tests applied to those examples sort of thing so that way I can become more familiar of when to use which test the most effectively
 
  • #11
For the comparison test you should choose a series that has a very similar form or a series that your are familiar with like the geometric series or the p-series. Also, when you have cos(n) or sin(n) use the fact that they are bounded.
 
  • #12
For your algorithm try drawing a binary tree. The root node should be the divergence test.
 
  • #13
so there's an algorithm to apply that always works?
 
  • #14
Try to come up with your own algorithm that always works for you.
 
  • #15
GreenPrint said:
so there's an algorithm to apply that always works?

Ummm, no. At least not in the sense that if A is an algorithm and you give it a series S as input it will always tell you whether or not it converges. What he means is that you need to actually solve some problems (yes, I know you don't want to; get over it and do it) and see which tests might apply to different series and which tests you should start with. For example, if you are solving an integral, you don't jump straight to trig. substitution or integration by parts, do you? Usually you just try to find an anti-derivative of the integrand, correct?

Same thing here. However, this requires you to not be lazy and actually SOLVE problems rather than looking at a list.
 
  • #16
Didn't we discuss this topic last month?

https://www.physicsforums.com/showthread.php?t=522400&highlight=convergence+tests

There is no Philosopher's Stone for this process. You have to work on a variety of series to learn which methods are more likely to work on a new problem, and which are pointless to try. Experience matters in developing the skill.

As for "gigantic lists of problems" and already worked-out tests, I have not found a source that has both (I know of plenty of books with long lists of series problems and just answers*...). You would need to look for (and study) a number of books and on-line sources that provide worked examples; I've yet to find a "Big Book of Infinite Series (with Convergence Tests!)"...

*I've even found the one "everyone" steals problems from...
 

FAQ: Calculus II - Determining if Infinite Sequences Converge

What is the purpose of determining if infinite sequences converge in Calculus II?

The purpose of determining if infinite sequences converge in Calculus II is to understand how a sequence of numbers approaches a specific value as the number of terms in the sequence increases. This information is important in many mathematical and scientific applications, as well as in understanding the behavior of functions and series.

How do you determine if an infinite sequence converges?

In Calculus II, an infinite sequence is said to converge if the terms of the sequence approach a specific limit as the number of terms increases. To determine if a sequence converges, you can use various tests such as the comparison test, the ratio test, or the integral test.

What is the difference between a convergent and a divergent infinite sequence?

A convergent infinite sequence approaches a specific limit as the number of terms increases, while a divergent sequence does not have a specific limit and may either approach infinity or oscillate between values. In other words, a convergent sequence has a finite limit, while a divergent sequence does not.

Can an infinite sequence converge to more than one limit?

No, an infinite sequence can only converge to one limit. If a sequence has multiple limits, it is considered a divergent sequence.

What is the significance of determining if an infinite sequence converges in real-world applications?

Determining if an infinite sequence converges has many real-world applications, including in finance, physics, and engineering. For example, in finance, the concept of compound interest involves an infinite sequence of compounding periods, and determining if this sequence converges can help predict future values of investments. In physics, the study of infinite series is important in understanding the behavior of electrical circuits and quantum mechanics. In engineering, the convergence of infinite sequences is used in the design and analysis of complex systems and structures.

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