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Introduction to Series - At A Glance:

The arithmetic series is the 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 and 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,...

is an arithmetic sequence.

Sample Problem

The series

1 + 3 + 6 + 10 + ...

is not an arithmetic series because

1, 3, 6, 10, ...

is not an arithmetic sequence.

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

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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