Notation for Series: Exploring the Difference

In summary, the first statement shows the sum of two terms while the second statement shows the sum of three terms. This is to make the pattern more clear to the reader and to avoid any confusion about the next term in the series. The ellipse (...) indicates that the series should continue in the obvious way, but it is ultimately up to the reader to interpret the writer's intentions. However, the convention is to give the first three terms of a series.
  • #1
Noxide
121
0
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...
 
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  • #2
Hi Noxide! :smile:
Noxide said:
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. :wink:
 
  • #3
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.
 
  • #4
JonF said:
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 + ...
 
  • #5
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.
 
  • #6
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.
 
  • #7
Rasalhague said:
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.
 

FAQ: Notation for Series: Exploring the Difference

What is the purpose of notation for series?

The purpose of notation for series is to provide a compact and efficient way to represent an infinite sequence of numbers. It allows us to easily manipulate and analyze these sequences, making it an important tool in mathematical and scientific calculations.

What is the difference between summation and product notation?

Summation notation (Σ) is used to represent the sum of a finite or infinite sequence, while product notation (Π) represents the product of a sequence. In summation notation, the numbers being added are placed inside the parentheses, while in product notation, they are placed next to the Π symbol. Additionally, summation notation uses a variable to represent the index of the sequence, while product notation uses the index itself.

How do you determine the difference between two series?

To determine the difference between two series, you subtract the terms of one series from the corresponding terms of the other. This can be done using summation notation, where the difference between two series is represented as Σ(an - bn), where an and bn are the terms of the respective series.

What is the purpose of exploring the difference between series?

Exploring the difference between series allows us to understand how changing the initial value or rate of change affects the overall behavior of the series. It can also help us identify patterns and make predictions about the series.

How can the difference between two series be used in real-world applications?

The difference between two series can be used in various real-world applications, such as analyzing stock market trends, predicting population growth, and calculating interest rates. It can also be used in physics and engineering to model and predict the behavior of complex systems.

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