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At a Glance - Arithmetic Series

The arithmetic series is one of the simplest series we can come up with. In terms of common restaurant menu items, the arithmetic series is a burger. Although it many be spiced up with bacon, feta cheese, and some type of questionable special sauce, it appears on every restaurant menu in one form or another. We need to understand this series type backward and forward.

An arithmetic series is a series whose terms form an arithmetic sequence. Simple enough, right? If we wrote down an arithmetic sequence, we could replace all of the commas with plus signs to get an arithmetic series.

Sample Problem

The series

1 + 3 + 5 + 7 + ...

is an arithmetic series because

1, 3, 5, 7,...

gets from one number to the next by adding 2 and only by adding 2. Because we are adding the same amount every time, this is an arithmetic series.

Sample Problem

The series

1 + 3 + 6 + 10 + ...

is not an arithmetic series because

1, 3, 6, 10, ...

gets from one number first by adding 2, then by adding 3, then by adding 4, etc. Because we're adding different amounts, this isn't an arithmetic series.

We have some good news. Just like a plain, old burger, we already know everything there is to know about the convergence of arithmetic series.

To start with, a constant series is an arithmetic series where the difference between successive terms is d = 0. The constant series

only converges if a = 0. That's a pretty bold statement. Take a look at a couple problems to understand why.

Example 1

Look at the arithmetic series

where ai = 4 + 3(i – 1)

Does this series converge or diverge?


Exercise 1

Look at the arithmetic series

where

ai = a1 + (i – 1)d

and both a1 and d are positive. Does this series converge or diverge?


Exercise 2

Look at the arithmetic series

where

ai = a1 + (i – 1)d

and d is negative. Does this series converge or diverge?


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