# Infinite Series

The sum of infinite terms that follow a rule.

When we have an infinite sequence of values:

12 , 14 , 18 , 116 , ...

which follow a rule (in this case each term is half the previous one),

and we add them all up:

12 + 14 + 18 + 116 + ... = S

we get an infinite series.

"Series" sounds like it is the list of numbers, but it is actually when we add them up.

(Note: The dots "..." mean "continuing on indefinitely")

## First Example

You might think it is impossible to work out the answer, but sometimes it can be done!

Using the example from above:

12 + 14 + 18 + 116 + ... = 1

And here is why:

(We also show a proof using Algebra below)

## Notation

We often use Sigma Notation for infinite series. Our example from above looks like:

 This symbol (called Sigma) means "sum up"

Try putting 1/2^n into the Sigma Calculator.

## Another Example

14 + 116 + 164 + 1256 + ... = 13

Each term is a quarter of the previous one, and the sum equals 1/3:

Of the 3 spaces (1, 2 and 3) only number 2 gets filled up, hence 1/3.

## Converge

Let's add the terms one at a time. When the "sum so far" approaches a finite value, the series is said to be "convergent":

Our first example:

12 + 14 + 18 + 116 + ...

 Term Sum so far 1/2 0.5 1/4 0.75 1/8 0.875 1/16 0.9375 1/32 0.96875 ... ...

The sums are heading towards 1, so this series is convergent.

In calculus we say "the sequence of partial sums has a finite limit."

## Diverge

If the sums do not converge, the series is said to diverge.

Example:

1 + 2 + 3 + 4 + ...

 Term Sum so far 1 1 2 3 3 6 4 10 5 15 ... ...

The sums are just getting larger and larger, not heading to any finite value.

It does not converge, so the series is divergent.

### Example: 1 − 1 + 1 − 1 + 1 ...

It goes up and down without settling towards some value, so it diverges.

## More Examples

### Arithmetic Series

When the difference between each term and the next is a constant, it is called an arithmetic series.

(The difference between each term is 2.)

### Geometric Series

When the ratio between each term and the next is a constant, it is called a geometric series.

Our first example from above is a geometric series:

(The ratio between each term is ½)

And, as promised, we can show you why that series equals 1 using Algebra:

First, we will call the whole sum "S":   S = 1/2 + 1/4 + 1/8 + 1/16 + ...
Next, divide S by 2:S/2 = 1/4 + 1/8 + 1/16 + 1/32 + ...

Now subtract them!

All the terms from 1/4 onwards cancel out.

And we get:S − S/2 = 1/2
Simplify: S/2 = 1/2
And so:S = 1

### Harmonic Series

This is the Harmonic Series:

It is divergent. How do we know? Let's compare it to another series:

 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + ... 2 3 4 5 6 7 8 9 etc... 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + ... 2 4 4 8 8 8 8 16

In each case, the top values are equal or greater than the bottom ones.

Now, let's add up the bottom groups:

 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ... + ... 2 4 4 8 8 8 8 16 1 + 1 + 1 + 1 + 1 + ... = ∞ 2 2 2 2

So our original series must also be infinite.

### Alternating Series

An example of an Alternating Series (based on the Harmonic Series above):

It moves up and down, but in this case converges on the natural logarithm of 2

To show WHY, first we start with a square of area 1, and then pair up the minus and plus fractions to show how they cut the area down to the area under the curve y=1/x between 1 and 2:

Using integral calculus (trust me) the area under that y=1/x curve is ln(2):

2
1
1/x dx = ln(2) − ln(1) = ln(2)

(As an interesting exercise, see if those rectangles really do make the area shown!)

## More

There are more types of Infinite Series, and it is interesting (and often challenging!) to work out if they are convergent or not, and what they may converge to.