Convergent or Divergent: Is this a Convergent Series?

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Discussion Overview

The discussion revolves around the classification of a series as either convergent or divergent, based on a lecturer's notes. Participants are examining the distinction between a sequence and a series, particularly in the context of a specific mathematical example.

Discussion Character

  • Homework-related
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants assert that the series in question is divergent, as stated by the lecturer, while others believe it is convergent.
  • One participant notes that the sequence converges to 2, raising questions about the implications for the series.
  • There is a clarification regarding the difference between a sequence (which converges to 2) and a series (which is suggested to diverge).
  • Participants express confusion regarding the terms "sequence" and "series," with one participant acknowledging a misunderstanding of the two concepts.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views on whether the series is convergent or divergent, with some clarifying the distinction between sequences and series.

Contextual Notes

There is a lack of clarity regarding the specific series being discussed, as well as the assumptions underlying the lecturer's classification. The mathematical steps leading to the conclusion of divergence are not fully resolved.

matthew1
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Mod note: Moved from a homework section.
1. Homework Statement

this is my lecturer's notes, he says it is a divergent series, but this seems like an obvious convergent series to me..
could someone verify?

Homework Equations



https://www.dropbox.com/s/mc5rth0cgm94reg/incorrect maths.png?dl=0

The Attempt at a Solution

 
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matthew1 said:

Homework Statement



this is my lecturer's notes, he says it is a divergent series, but this seems like an obvious convergent series to me..
could someone verify?

Homework Equations



https://www.dropbox.com/s/mc5rth0cgm94reg/incorrect maths.png?dl=0

The Attempt at a Solution

The series is the sum of all of the terms of the sequence. The sequence does converge to 2.

What is the sum of an infinite number of terms each of which is at least a little greater than 2 ?
 
SammyS said:
The series is the sum of all of the terms of the sequence. The sequence does converge to 2.

What is the sum of an infinite number of terms each of which is at least a little greater than 2 ?
Ah, I thought i was already looking at the series, but that was the terms of the sequence! thanks a lot :-)
 
matthew1 said:
Ah, I thought i was already looking at the series, but that was the terms of the sequence! thanks a lot :-)
Be sure you learn the difference between a sequence of terms, such as ##\{2 + e^{-m}\}_{m = 1}^{\infty}##, and a series, such as ##\sum_{m = 1}^{\infty}2 + e^{-m}##.
 

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