Is this series convergent or divergent?

[matthew1]

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

 

[SammyS]

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 ?

 

[matthew1]

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 :-)

 

[mathman]

please tell me this is wrong? this is my lecturer's notes but this seems like a convergent series to me..

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

You are confusing series and sequence. As a sequence (individual terms) it converges to 2. As a series (sum of terms) it diverges.

 

[Mark44]

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 $\sum_{m = 1}^{\infty}2 + e^{-m}$.