Determining convergence/divergence

  • Context: MHB 
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Discussion Overview

The discussion centers around methods for determining the convergence or divergence of sequences and series, particularly in the context of preparing for a test. Participants explore various approaches and tests that could be employed, including the use of calculators.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about the easiest way to determine convergence or divergence, expressing urgency due to an upcoming test.
  • Another participant suggests that if the limit of a sequence approaches a real number, then the sequence converges, but notes that oscillating sequences may diverge.
  • A different participant emphasizes the need for an efficient test for convergence or divergence, recommending the Raabe test for series with positive terms.
  • There is a request for clarification on what constitutes 'easy' or 'quickly' in the context of determining convergence.

Areas of Agreement / Disagreement

Participants express different perspectives on what methods are considered easy or efficient for determining convergence or divergence, indicating that there is no consensus on a singular approach.

Contextual Notes

Participants have not defined specific conditions under which their suggested methods apply, nor have they resolved the effectiveness of different tests for various types of sequences or series.

Who May Find This Useful

Students preparing for tests in calculus or mathematical analysis, particularly those focusing on sequences and series.

ineedhelpnow
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what's the easiest way to quickly determine convergence or divergence? or is there a way to do it by calculator? i have a test today and most of the questions ill be given a series (few sequences) and itll be asking whether they converge or diverge?
 
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If you can show that:

$$\lim_{n\to\infty}a_n=L$$

where $L\in\mathbb{R}$, then the sequence converges. If you doubt the outcome of your limit, then you could evaluate $a_n$ for large values of $n$ as a means to try to verify your result. Just be mindful that if $a_n$ contains a power of $-1$, then the sequence could oscillate between two limiting values and is thus divergent.
 
ineedhelpnow said:
what's the easiest way to quickly determine convergence or divergence? or is there a way to do it by calculator? i have a test today and most of the questions ill be given a series (few sequences) and itll be asking whether they converge or diverge?

What do you mean by 'easy' or 'quickly'? ... in Your place I would look for the most efficient test to determine if a series converges or diverges ... for the series with positive terms the more efficient test in my opinion is the Raabe test...

Kind regards

$\chi$ $\sigma$
 
@chisigma i mean the simplest method that can help me get through the problems quickly
 

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