Calculus II - Series and Convergence

In summary, the conversation discusses determining if the series Σn/(2n+1) converges, using the divergence test. The attempt at a solution initially thought the series diverged, but since the limit of the summand was found to be 1/2, it was determined that the series actually converges. However, it is important to note that the converse is not always true and different methods may be needed to determine convergence.
  • #1
GreenPrint
1,196
0

Homework Statement



Determine if the series

inf
Sigma n/(2n+1)
n=1

converges

Homework Equations





The Attempt at a Solution



When i did this I originally I thought I would just apply the divergence test

lim n/(2n+1) =/= 0
n->inf

there fore I thought by the divergence test the series diverges but I guess sense the limit is defined to be 1/2 it converges...

I'm confused... I think I may be over complicating this but by the divergence test shouldn't this series diverge sense the limit does not equal zero?

Thanks for any help
 
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  • #2
GreenPrint said:
... but by the divergence test shouldn't this series diverge sense the limit does not equal zero?

Yes, that's correct. It's more commonly called the nth term test for divergence.
 
  • #3
GreenPrint said:

Homework Statement



Determine if the series

inf
Sigma n/(2n+1)
n=1

converges

Homework Equations





The Attempt at a Solution



When i did this I originally I thought I would just apply the divergence test

lim n/(2n+1) =/= 0
n->inf

there fore I thought by the divergence test the series diverges but I guess sense the limit is defined to be 1/2 it converges...

I'm confused... I think I may be over complicating this but by the divergence test shouldn't this series diverge sense the limit does not equal zero?
Thanks for any help

The bolded part is correct. Since you took the limit of the summand and received 1/2, the limit diverges. If the sum converges its limit is equal to 0.

However, the converse of the statement is not true. Don't fall into the trap of thinking the sum converges if the limit is equal to 0. If you obtain a limit of 0, you need to resort to a different method to determine if the sum converges.
 
  • #4
hm thanks i think i got it mixed up and it asked me if the sequence converges which it does lol
 
  • #5
GreenPrint said:
hm thanks i think i got it mixed up and it asked me if the sequence converges which it does lol

No it doesn't. It diverges. You just showed this using your test. Or was that a typo?

And it's a series, not a sequence.
 
  • #6
ya i got it and understand now i got a different part mixed up and stuff
 

FAQ: Calculus II - Series and Convergence

What is Calculus II - Series and Convergence?

Calculus II - Series and Convergence is the second course in the calculus sequence, focusing on the study of infinite sequences and series. It builds upon the concepts learned in Calculus I and applies them to more complex functions.

What topics are typically covered in Calculus II - Series and Convergence?

The course covers topics such as convergence and divergence of sequences and series, power series, Taylor and Maclaurin series, and tests for convergence such as the integral test, comparison test, and root test. It also includes applications of series and convergence to real-world problems.

Why is studying series and convergence important?

Series and convergence are important in many fields of mathematics and science, as they provide a way to approximate functions and solve complex problems. They are also used in many practical applications, such as engineering, physics, and finance.

What are some common challenges in learning Calculus II - Series and Convergence?

One common challenge is understanding the concept of convergence and how to determine whether a series converges or diverges. Another challenge is learning the various tests for convergence and when to apply them. Many students also struggle with the complex algebra and manipulation involved in solving series problems.

How can I prepare for Calculus II - Series and Convergence?

Preparing for Calculus II - Series and Convergence involves reviewing the material covered in Calculus I, particularly limits, derivatives, and integrals. It is also helpful to practice solving series problems and familiarize yourself with the different tests for convergence. Seeking help from a tutor or attending extra review sessions can also be beneficial.

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