A series is an infinite sum of the terms of sequence.

we can ask two important questions about it.

- Do the numbers in the list approach a finite value?
- Can I sum all of the numbers in the list and obtain a finite result?

The first question is really whether the *limit* exists and we studied several ways to
determine this previously. As it turns out, the second question is closely related to
the first one. We motivate this through an example.

Suppose that we want to study the infinite sum below.

A student, feeling quite clever, decides to illustrate the sum by drawing a square with side length one unit and shading it in a special way. In order to understand the concepts here more fully, it is recommended that you draw and shade as you read.

Step 1: Shade the left half of the square.

- Call the area shaded this step . Then, we have .
- Call and the total shaded area of the square . Then, we have .

Step 2: Shade the bottom half of the unshaded region.

- Call the area shaded this step . Then, we have .
- Call and the total shaded area of the square . Then, we have .

Visually, notice that we can find by noting and writing down the total shaded are of the square.

Analytically, we can write

Step 3: Shade the left half of the unshaded region.

- Call the area shaded this step . Then, we have .
- Call and the total shaded area of the square . Then, we have .

We can think of visually or analytically.

Hopefully, the pattern used to shade the square is becoming clear. We can define the area shaded during the -th step to be , and can observe that .

We can also let denote the total shaded area after the -th step. Analytically, we have , or by using summation notation, we can write .

Looking at the pictures drawn so far, notice that the only unshaded area after the -th step is a rectangle of area , so we can write a formula for .

We can now evaluate the limit and find that .

We also have another method of thinking about this limit; after we continue shading the square indefinitely, there will be no portion of it that has not been shaded. Thus, the total shaded area should be .

We would thus like to conclude that

### Infinite series

The above scenario can be modeled using sequences. We have whose -th term is given by the explicit formula , and we represent the sequence by the ordered list below.

and we can interpret the series as the attempt to add up all of the terms in the above sequence. In order to determine if this can be done, we found a new sequence that was created from the original sequence. It is represented by the list below.

Note that the *limit* of this new sequence is exactly the *sum* of all of the terms
in the old sequence! Let’s formalize the ideas in the last example with a
definition.

**sequence of partial sums**of .

- (a)
- The series
**converges**if and only if exists. Furthermore, if , we say the series converges to . - (b)
- The series
**diverges**if and only if or otherwise does not exist.

The above definition really is assuring us that the symbols and are exactly the same! However, this definition makes the content of the previous example more precise. The major idea here is that we have techniques that we can use to determine whether limits exist and can even find what those limits are sometimes. Since we are now able to recast the new question “Can I sum all of the terms in a sequence?” into the old question “Does a sequence have a limit?”, we can now utilize all of our previous techniques to analyze the sequence of partial sums.

Now, let’s see an example.

Determine whether converges or diverges. If it converges, give its value.

### Properties of sums

We finish this section by giving some properties of series.

*convergent*series and suppose that and

- (a)
- Constant Multiple Rule
For any constant ,

- (b)
- Sum/Difference Rule

Notice, of course, that we’re working with convergent series in this theorem. Adding divergent series is trickier, but there is something we can say about the attempt to add a convergent series and a divergent series.

To understand why this theorem holds, note that if would converge, by the last theorem, we would know that the difference

converges since both series above are convergent. Furthermore, the previous theorem also guarantees that

but this is precisely , which diverges by assumption. This is a contradiction, so must diverge.

Essentially, this theorem ensures that it does not matter where we *start* summing
the terms of a sequence. Since we have infinitely many terms to try to add
together, the sum of the first finitely many will not affect *if * this addition is
possible.

We now finish the section with an example that ties many ideas together.

Let’s write out the series and make a few observations.

Putting this together, we have:

and we find that .

We can determine whether converges or diverges by analyzing . Since this limit is zero, we know that converges to .