What Are the Key Differences Between These Summation Notations?

[Noxide]

Please explain the difference between these two statements.

1 + 2 + ... + n = 1/2n(n+1) for all n in the natural numbers

1 + 1/2 + 1/4 + ... + 1/2^n = 2 - 1/2^n for all n in the natural numbers


Why does the first explicitly show two terms being summed whereas the second shows 3 terms being summed...

I don't think I have a good understanding of how to work with these things...

 

[tiny-tim]

Hi Noxide!

Why does the first explicitly show two terms being summed whereas the second shows 3 terms being summed...

No particular reason …

the writer just puts in as many terms as he thinks makes it clear what he means.

 

[JonF]

Just to expand on what tiny tim said
1/1 + 1/2 + … doesn’t establish a clear pattern:

It could mean 1/1 + 1/2 + 1/3 + 1/4 + …
or 1/1 + 1/2 + 1/3! + 1/4! + …
or 1/1 + 1/2 + 1/2^2 + 1/2^3 + …

It’s always helpful to give the reader a definite idea of what the series means before the ellipse (…) and the general term.

 

[Rasalhague]

Just to expand on what tiny tim said
1/1 + 1/2 + … doesn’t establish a clear pattern:

It seems no less clear than 1 + 2 + ...

 

[zcd]

One would most likely guess that 3 comes after 1 and 2, which is the writer's intent. For the geometric series 1/2^n however the writer wants to make sure the reader doesn't guess 1/3 for the next term and provides 1/4 instead.

 

[TylerH]

The ellipse means to continue in the obvious way. It's ultimately up to the reader to decide the clarity of the intents of the writer. The convention is to give the first three terms, but there are exceptions.

 

[Rasalhague]

It seems no less clear than 1 + 2 + ...

Oh, sorry, I wan't paying attention! Yes, 1+2+... the most obvious guess is 3. But in the second example, the next term is not the most obvious 1/3, but rather 1/4.